It seems unlikely. The legal thinking throughout the war was that the Union was indivisible. The states in rebellion had never actually left and so would not be subject to being denied statehood. A military occupation as seen IOTL is the most likely outcome.
Few points. Foster's first action on becoming Acting POTUS will probably be to recall Congress, if only to secure the election of a House Speaker to be next in line of succession after him. So we may well get the 14th Amendment a year ahead of OTL. And Foster, instead of egging on the South to reject it (as Johnson did) will presumably be twisting arms in *favour* of ratification. Given that at this point the South is still in a state of shock post-Appomattox. the Rebel States may well ratify, in which case they may be readmitted without Black suffrage (except maybe for the literate and Union veterans). which makes for a *very* different Reconstruction.
Minor point. Under the Act of 1792 a Presidential election should be held in Nov 1865. However this would put that and subsequent elections out of synch with Congressional ones, which many pols might not want*. So might Foster's interim Presidency be extended for another year in order to avoid this? After all if he is getting on reasonably well with Congress there is no urgent reason to replace him.**
*It is true that in those days many states had elected their Congressmen in the odd-numbered year, but iirc, by 1865 only two *Northern* states, CT and RI, still did this. and of course the South wouldn't be voting on the matter.
**If indeed they replaced him at all. If they don't have Grant, then the Acting POTUS who had seen them through the crisis might seem as good a choice as any.