There is a subtle issue here but it is not where the OP thinks it is. Any explicitly written integer is obviously "standard" whereas each new integer arising in the ultrapower of $\mathbb N$ is obviously "nonstandard". The subtle issue is that in the particular $\mathbb N$ we are working with, there may be integers (certainly bigger than the explicitly written ones) that look nonstandard from the viewpoint of another set-theoretic universe.
But this does not change the fact that any concrete proof has "standard" length. And since we are accepting ZFC axiomatics for each instance of a universe in Hamkins' multiverse, this has nothing to do with ultrafinitism.
In this publication (see also here) we proved a theorem that each universe in the Hamkins-Gitman "baby model" actually turns out to be a model of a variant of Edward Nelson's Internal Set Theory. Translated into the terms of Robinson's framework, this means that the integers in each instance of a universe in the multiverse contain not only an initial cut including all explicitly specifiable integers, but also a wealth of integers that are infinite according to a "smaller" universe in the multiverse.