Did Biden ask the State Department to lift Kolomoyskyi’s travel ban?

Offering an alternative explanation to the one already given. I came across this as part of my own research, so take this with a grain of salt (although I certainly don't buy into the fluffy "rabbi" explanation).

In this Harper's Magazine article, Matthew Rojansky, director of the Kennan Institute at the Woodrow Wilson Center for International Scholars is quoted by the author:

Poroshenko [...] enlisted ambassador Pyatt, [...] in a deal to remove his rival from the scene. “My understanding is that part of the deal whereby Kolomoisky gave up his attempt to take over control of Ukrnafta and UkrTransNafta and gave up governorship of Dnipropetrovsk and gave up having his pawn in control of Odessa,” Rojansky told me, “was that the U.S. ambassador came in as an intermediary guarantor and said if you do these things, we will take you off the visa bad list.”

So we have at least one source, Rojanksy - who seems to be well-respected - with the understanding that the State Department (in the form of Pyatt) was directly involved in lifting the ban, apparently at the request of Poroshenko.

We can only speculate as to whether or not Poroshenko had in turn spoken directly to Biden about this request as well. None of the leaked phone calls between Biden and Poroshenko have anything about it (at least not the ones we have heard so far).

Circumstantial at best. We have this from Tablet Magazie that details the specifics of the lifting of the travel ban. While theoretically the direction could have come from the Secretary of State or Joe Biden, such involvement was a complete omission from those that were close to the affair. Here a Reformer, who would be antagonistic to perpetuating corruption (although he could be more concerned about Ukrainian corruption, makes the case that the deal was between the Ukrainian US embassy and a third party.

The news of his departure from Ukraine—and that the American embassy in Kiev had relented on the oligarch’s longstanding visa ban—was delivered over Facebook by reformist member of the Rada (Ukraine’s Parliament), Sergey Leschenko. Leschenko was one of the idealistic young MPs who fought against Kolomoisky’s power inside the Rada. His post, translated into English below, was understandably euphoric:

News on my favourite topic. Kolomoisky took advantage of a recent conversation with the US ambassador in order to expiate himself of his American visa ban. This was brokered by the rabbi of Dnipropetrovsk Kaminetsky, who is an American citizen. A special exception was made, and Kolomoisky was issued a short stay, single entry visa for the US, where he is now mostly devoting his spare time to attending basketball games. Kolomoiskiy’s family members, were in contrast, given ten-year visas. However, sources do not specify whether Kolomoisky had the visa placed in his Ukrainian, Israeli or Cypriot passports.