Despite the fact that I have no time for this right now, I was so curious to know the answer to this that I had to do some research!

It's surprising, though, that kosher observance is the reason your relative evaded the army, as the Austro-Hungarian Army had a strong culture of religious Jewish participation. As a general rule, the Austro-Hungarian Army was reasonably accommodating to Jewish soldiers, allowing them leave to hold religious services such as Passover seders; there were many devoutly religious soldiers who, though they had fewer food options than their non-kosher-observing compatriots, were still dedicated soldiers while also being dedicated Jews who brought prayer books, Torah scrolls, prayer shawls, and other ritual items to the battlefield (though not necessarily enough for all who wanted them) and who often organized daily prayers. There were also a large number of Jewish chaplains as well as volunteer chaplains , rabbis who lived near army bases who would visit to assist the soldiers. Usually, the many Jewish soldiers who wanted to keep kosher would create their own field kitchens, slaughter their own animals, cook their own vegetables on kosher utensils, and/or eat only packaged items like bread, butter and jam. If he'd stuck around, though, then by 1916 kosher kitchens would have been ordered to be supplied in any unit with more than 100 Jews, allowing for kosher food at all times, though it's unclear if any of these kitchens were ever actually set up.

So basically, observant Jewish soldiers in Austro-Hungary had it pretty good- the devout were often so secure in their religious observance that some would even request from local Jewish communities specific kinds of prayer books, not just the standard one carried by soldiers! So I find your relative's story super interesting in that regard- though it's still very possible that he did not feel that his religious observance would be able to be sufficient in the army.

Now, of course the war came to the US long after it did to Austro-Hungary, and Jews who faced the draft were very concerned about whether they would have the ability to keep kosher, keep the Sabbath, celebrate holidays, have Jewish chaplains, etc. The Orthodox Jewish Yiddish papers in New York even advocated for special Jewish regiments along the lines of the recently formed Jewish Legion in the British Army; this was however seen as a terrible idea by more assimilationist-minded Jews who wanted to show Jewish army service as an example of the way that Jews were just like any other American citizens (especially due to the fact that Jews were being targeted by accusations of draft-dodging and shirking of duty- despite the fact that they more than filled their neighborhood quotas- which would continue to plague them in WWII).

An organization called the Jewish Welfare Board was soon created to initiate a Jewish infrastructure in the army by preparing religious materials, recruiting potential chaplains, and lobbying the US government to provide kosher food, though these efforts weren't necessarily seen as sufficient by the more religiously conservative. They were funded by vast fundraising efforts by the New York Jewish community. While the War Department recognized the JWB as an official organization, there were still relatively few services offered, especially initially- the first observant Jewish soldiers quartered at Camp Upton (of whom there were nearly 2500), for example, had chaplains, no kosher food but what was sent by their families, and few organized religious services, either Reform or Orthodox (in particular, no Orthodox rabbis could come to perform Sabbath or holiday services as they could not be driven to the camp). However, this soon changed, as guest quarters were built for visiting rabbis and Torah scrolls were placed in the Interdenomination Chapel at Upton. While the soldiers were furloughed for Yom Kippur, they had on-site services for Rosh Hashana, with a rabbi brought in specially due to the lack of chaplains.

While the US Army was generally very inclusive of Jews in their development of the chaplaincy in WWI, there were never more than a dozen Jewish chaplains, and most Jewish services were provided by the JWB, which considered itself Jewishly neutral- trying to provide for Reform, Conservative and Orthodox Jews in one fell swoop (and often satisfying nobody). Unfortunately (and surprisingly), I haven't been able to find any information about whether kosher food was provided to Jewish soldiers overseas- if it was, it was probably through private organizations like the JWB, through donations from Jewish civilians, as it had been at Camp Upton. So ironically, it's very probable that it was the Austro-Hungarian Empire (due to its more established infrastructure) that provided a somewhat more Jewish-friendly environment than the United States!

Sources:

Rozenblit, Reconstructing a National Identity: The Jews of Habsburg Austria During World War I

Sarna, American Judaism: A History

Sterba, Good Americans: Italian and Jewish Immigrants During the First World War