No, NO animal not recorded as 100% Arab is ever allowed to be registered as Arab, it does not matter how many generations of purebred you have, if there is one that is not registered it is not allowed.
Every single Arab registered today can trace it's lineage to the desert and thereon to verbal pedigree.
The mare is PUREBRED no Pinto except what was already there in the "high whites" we have all seen for many years.
You have NO way of knowing what is in your stallions past, Pinto wise, unless you know his ancestors by pictures or hearsay (ie someone who has seen them.
If you are relying on the AMHA or even the AMHR to give you the correct colour or pattern you are foolish to place such reliance.
It is true that, before DNA, someone could have slipped a Pinto into the Arab genes- but why would they have done??
It is only recently that Pinto purebreds have been allowed to be registered- since DNA testing, in fact, as, before that, even when they really were "pure" they were discarded!!
Anyway the mare's pattern is consistent with Sabino, without Tobiano, so I have no problem with it.
And, Yes, your stallion could sire a Pintaloosa- you would probably not recognise it as such but, yes, it could happen.
Has he had blaze faced foals???
If so, there is your (minimal) Pintaloosa.