Expanded from an earlier answer of mine

The process of coaling was an arduous and backbreaking task in the early twentieth century. This was doubly true for the fleets commanded by Zinovy Rozhestvensky and Dmitry von Fölkersam which had to make a long journey to Asian waters. The journey was a logistical nightmare for a Russian navy that was largely designed to fight in European waters. The inexperience of the fleet, coupled with panic over Japanese sneak attack led to various scares during the travel that the fleet was under attack. But the Russians were able to take advantage of the emerging international maritime law that allowed belligerent warships to coal and take on water in neutral ports.

The Russian fleet mostly sought shelter in friendly French ports like Dakar, Madagascar, and Camn Ranh Bay. International maritime law allowed for belligerent warships to stay 24 hours in neutral ports for coaling, but the Franco-Russian alliance meant that the French officials often turned a blind eye to the Russians overstaying their limit, much to the chagrin of Japanese diplomats. The Russians also contracted out to private French and German civilian companies for colliers. In the latter case, the Hamburg-Amerika line's colliers further strained Anglo-German relations, and Chancellor von Bülow wanted the company to cancel the contract, but Kaiser Wilhelm II sided with Hamburg-Amerika as it fit into his conception of using German support for Russia as a wedge against the Franco-Russian alliance. In this instance, the French desire to keep the alliance alive and the German desire to undermine it actually both worked in the Russians' favor, making the fleet's voyage possible after the Dogger Bank incident alienated an already cold Britain.

Rear Admiral Fölkersam's newly-christened Third Pacific Squadron though had a slightly-shorter trip than the main force as it made the transit via the Canal. One rationale was that Fölkersam's ships were both lighter and unable to carry as much coal as the main battle line under Rozhestvenskii's flag. There was a fear that these smaller ships would hamper the already complicated logistics of the voyage. Moreover, there was also a fear that the heavily-laden battleline might also find itself aground in the narrow waters off Suez because the amount of coal taken aboard had increased their draft.

But it was more than just practical matters of logistics that underlay the division of the fleet. The same fears that inspired the Dogger Bank incident were still at play and there was a fear that the Japanese could have naval forces in the restricted waters of the Mediterranean. Fölkersam's ships were not only a tad more maneuverable, they were also expendable for the whole fleet. His ships were either older ones or auxiliaries.

The expendable nature of Third Pacific Squadron also reflected the fear that Britain could prevent the whole enterprise. Rozhestvensky was afraid that the British control over the Canal meant they could use whatever pretext to detain or even seize the battlefleet. The 1888 Convention of Constantinople sanctified the Canal as an international waterway and guaranteed rights of access. But the Convention also had provisions governing the transit of warships and ships of belligerent powers that limited their loiter times in the Canal or coaling at Canal ports. The British, as the de facto controlling power of the Canal, had the power to interpret the Convention's provisions. They had already done so during the Spanish-American War when they narrowly interpreted the Convention to minimize the time Spain's warships spent coaling and refitting while making a transit to the Philippines. The Dogger Bank incident certainly did not endear the Russian fleet to the British and there was a fear that the British could interpret the Convention against Russia's interests. Moreover, the Convention was still a relatively untested treaty and it was unclear from Rozhestvensky's perspective whether or not the British would abide by it or if the Egyptian government would alter the rules for coaling.

So even if Dogger Bank did not alter the Convention, the bad blood the incident created between Britain and Russia had the potential for British to use the Convention against Russia. This, coupled with the already severe logistical problems of the fleet, meant that sending Fölkersam's more expendable and lighter ships to transit the Canal was a calculated risk on the part of Rozhestvensky and one that paid off. The Third Pacific Squadron was able to rendezvous with its colliers and then link up with the main force after the Canal transit. The British did not unduly delay or pester Fölkersam's detachment and the transit showed Britain's commitment to the Convention. Unfortunately, the subsequent Japanese triumph at Tsushima negated the herculean effort that had gone into supplying the fleet for its transoceanic voyage.