It’s not clear to me in the OP if this situation is a matter of the lady wanting to join the Roman Rite or if she wants to remain Byzantine. Two very different things.
(1) If the former, I suspect the local Latin pastor simply wants to know which Byzantine jurisdiction to contact to arrange the transfer. I’m not a canon lawyer, but I seem to think that in the absence of a local jurisdiction, the “old” rule (i.e., through Rome) would still apply.
(2) If the latter, there would appear to be 3 possible scenarios:
(a) this is a matter of the lady simply being received into the Catholic Church and geographically staying where she is, without effecting a “change of rite.” (The assumption here being that she attends the Latin Rite church in the first place because there is no Orthodox or ECC church in the area. In other words, if there were an EOC or ECC local, she would already be at one or the other.) If this is the case, the “transfer” would probably be from Serbian Orthodox to the Apostolic Exarchate of Serbia and Montenegro, which means she would remain status-quo in the US. Absent a Serbian jurisdiction in the US, (or another Byzantine jurisdiction that has been given official charge of the Serbians), she would come under the local Latin Ordinary. Here again I suspect that the request would have to go through Rome for the same reason as above.
(b) if this is a matter of choice, i.e., where would the lady be most comfortable, there would seem to be options. (The assumption here is either as in (a) above or that she has already investigated and dismissed locally available EOC options.) I’m not an expert on Byzantine usages, but from the standpoint of similarities, of those Churches with a jurisdiction in the US I think the Romanian would probably be closest to Serbian. Of course language would present a big difference. There are, I believe, also a handful of Albanian churches in the US, (presumably under the local Latin ordinary) but again language would be an issue. Language would be far less a problem with the Bulgarian, but I don’t know if they have a presence in the US.
(c) if this is simply a matter of determining which Byzantine jurisdiction the lady would come under, (and again we have the same assumption as in (a) above), it depends on whether any of the Byzantine jurisdictions in the US have been given official charge of the Serbians. If there is one, that answers the question. If not, I think it has to revert to (a) or (b) above.
FWIW, my guess is that scenario 1 and scenario 2a above are the most likely.