It really wasn't much, if at all, because the only two countries to come close to entering the war were Britain and France, and they would've entered on the same side. So, even if they had, there would not have been another puppetmaster on the other side (aka with the United States). Any attempt to construct what the US-side would have looked like, I would argue, veers too close to "what-ifs."

But even a US vs. Rebels-Britain-France coalition dies fairly early for a few reasons. The South was relying (too much) on the idea that "Cotton Diplomacy" would float them: Britain, France, and others would want favorable trading terms for cotton and would thus assist the Confederacy in some way. Ideally for them, Britain and/or France would enter the war directly. As I'll discuss later, this did almost become a real possibility. But without a certain diplomatic kerfuffle that happens in early 1862, I would argue this would not have become a particularly likely possibility. Britain was at the time worried about its fading influence on the continent, and France was eyeing down Mexico believe it or not. But the question is whether these interests would have been enough to enter the war at any point as it happened, and I would argue no.

Cotton Diplomacy itself fails to attract sufficient interest in part due to burgeoning colonialism, or rather the changing face thereof. Africa and Asia were by this point bonanzas. Though not alone, Britain and France are at that time carving up the world in places where they have fewer moral qualms (i.e. about treatment of "savage" peoples) and the natural resources are abundant and largely accessible. And, most importantly for Britain, Egypt and India become hugely successful cotton producers. Indeed, the tenuousness of cotton production and sale in the areas of the US in rebellion births cotton industries the world over. So Britain, more likely to enter in any serious way anyway, looks askance at Cotton Diplomacy. This is, of course, all the more exacerbated by the South's declining performance as the war drags on.

For one, the Confederacy had long been gambling on the notion that they would merely have to fight a defensive war, giving them favorable conditions and incurring considerably less human, material, and financial costs. What they didn't count on was that Lincoln would be so astute in his handling of any number of things, perhaps most importantly the Border States. Maryland was put under Marshall Law because it was the closest to insurrection, while soldiers were marched into Missouri but told not to do anything (i.e. just show our muscles). Delaware is the most loyal Border State, by contrast, and it is largely treated with kid gloves. Basically, Lincoln was handling each situation carefully and treating them as unique. The Emancipation Proclamation also famously excludes slaves in Border States from being freed by the decree. (Note: this is why Juneteenth only tells part of the story: slavery was only ended in the Confederacy on June 19th, 1865. Slavery persisted in the US as a whole until the ratification of the 13th amendment later that year.) Which is all to say that the South lost any hope of participation in earnest from those states.

Another hugely important blow to the South's viability as an ally for foreign powers is the capture of New Orleans, mouth to the all-important Mississippi River, fairly early in 1862. Southern strategists had prepared far more effectively for an attack down the Mississippi then up from the Gulf of Mexico. Taking New Orleans meant cutting off perhaps the most important artery in the South.

So Britain and France pretty quickly realized that things were not going so hot for the Confederacy's rebellion.

But I've gotten this far without discussing the closest the great foreign powers came to being involved in the war, because it was actually a possibility at one point; enter the Trent Affair.

In late 1861 a Union battleship stopped a British mail vessel and arrested two Confederate envoys headed to England to make their case for Britain's entering the war. Britain is immediately outraged because they argue it violates international laws that would protect (what are essentially) diplomats aboard a neutral ship. Britain gets super pissed off and actually begins mobilizing for the possibility of war, and France says they will follow Britain's lead in the conflict.

Lincoln, astute politician he is, relents and frees the two men, allowing them safe passage to Britain and thus on to other European powers thereafter. Britain backs down from agreeing to back the rebellion, however, and no one else takes the bait. And thus a major flicker of hope for their ramshackle rebellion dies.

Now, to give one more side to this story, you do have very concerted governmental and non-governmental efforts to avoid (especially) Britain's entrance into the war. Centrally, the US was trying to prevent naval assistance (from the world's greatest naval power) to the fledgling (and eventually, foundering) rebels. In any guise it did take, this assistance was in the form of shipbuilding and patently not in the use of actual British ships to fight the war.

So the US government is doing its damnedest to lobby Britain against its entering the war, but you also have heavy, heavy involvement from abolitionists. Abolitionism was quite popular, believe it or not, among many wealthy US-Americans and Britons, perhaps most famously in Quakers, whose religious movement is essentially bipolar at that time between the US (and to a lesser extent, Canada) and the British Isles (especially England). The Cadbury chocolatiers are perhaps the most famous example of wealthy Quaker business(mostly)men who participated in international networks of pacifists and, thus, abolitionists. These (mostly, but not exclusively) men are putting as much pressure as they can on prominent British politicians and wealthy and/or landed Brits to avoid its entering the war.

So the long and the short of it is that, no, there was no proxy war. The war veered (to some degree, the precise measure of which is still heavily argued) close at one point to major foreign powers entering a domestic conflict. Even still, there is no clear historical evidence to support the notion that any power would join the US directly to oppose the English and the French, and any attempt, I would argue, to name such a power would be too much of a what-if scenario for this subreddit. The US is in a weird position at that moment: fledgling and in a crisis of identity--a teenage country if there ever were one. It comes out of the conflict and quite stridently accelerates its efforts at modernization.

It's possible I'll expand on this, but for now, suffice it to say that the South had, to this point, stymied efforts we now associate with modernization--consolidation of federal power, investment in national-level infrastructure, investment in education, etc. For a brief period during the war and especially after a backlash against Johnson, these efforts are unleashed and a period of explosive modernization takes hold in the US. The US matures as a great power--including, unfortunately, through its maturation as a colonial power--as the century draws to a close, and it enters the twentieth century on wildly different footing than it had the nineteenth.

Sources:

Bacon, Margaret Hope. The Quiet Rebels: The Story of the Quakers in America. Philadelphia: New Society Publishers, 1985.

Caroll, Francis M. "Diplomats and the Civil War at Sea." Canadian Review of American Studies 40, no. 1 (2010): 117-28.

Griffin, Matthew. "George Thompson, Transatlantic Abolitionism, and Britain in the American Civil War." Slavery & Abolition 40, no. 3 (2019): 563-82.

Sheehan-Dean, Aaron, ed. A Companion to the Civil War. Malden, MA: John Wiley & Sons, 2014.