I generally agree with you that the language in Rumi and Walad's poems is very underwhelming. It does not appear to be of any (identifiable) Greek dialect and the list of alleged Cappadocianisms is very poor, in my opinion. Of course, the texts are from a very early period and one would (rationally, I think) expect Cappadocian and the other Asia Minor Greek varieties to start to diverge at that period at the earliest and then even more intensively after the fall of every kind of Greek power centre in the region, be it the Byzantine Empire (or the pathetic sod of what was left of it) or the Trabzon Empire.
Gregoire's list of Cappadocianisms is a joke: θέκνω is as good as it gets. But I indeed find it hard to believe Rumi picked up a Constantinopolitan Standard Greek that was distinct from what was spoken in Iconium. The use of σκήνωμα means that there was (inevitable) learned influence even here, but what they are presenting is not learned Greek.
Yet, I find some things to be potential indications of dialectal speech such as the στο σον το χείλο and the ση prepositional phrase (from σε + τη) in the 'ση εστία μου'. However, why does this Pontic στ > σ change appear only there but not in στο σον το χείλο which would have to be σο σον το χείλο or, to be more pedantic, σο σον σο χείλο?
Dunno if I'd trust that instance of ση, for that reason. I don't want to generate a new edition, but the Persian script is just tydhs as astya mw; and given Walad's allergy to articles (a foreignerism, I suspect), I'd just render this as τι είδες εις εστία μου.
I'm also thinking the lexical peculiarities (e.g. πορπατώ), how sure can we be of them? I haven't looked into the history of the word in the various dialects and periods, but for some reason I feel less keen to accept such examples as evidence of dialectal features in the text. Of course, the Arabic script poses many problems and I think I personally would stick to the uncontroversial ones which are somewhat interesting.
The Arabic script is extremely problematic, and the scribes had no idea what was going on; what'd be interesting to work out is, whether scribal errors can be explained by soundalikes as well as lookalike letters (i.e. internal dictation). Recall also that B & M had to basically make a new edition of Walad, seeking out a couple of manuscripts made the most sense; but this is still haphazard stuff. So no, I wouldn't trust πορπατώ... except that the Rababname has vowel pointing, and here reads straightforwardly powrpaty. Hm. This'd at least be worth checking whether πορπατώ turns up this far east....
An interesting example is the accusative plural άλλους in the verse το πωρικό το πικρό δος το άλλους. It is very suspicious and reminiscent of the Cypriot Greek genitive/accusative syncretism in the plural of the masculines (CG το φαΐν τους σιύλλους instead of το φαΐ των σκύλων or something like that). I have also found some similar examples in my Cappadocian texts. Can you see a pattern emerging there? That, of course, is if and only if the Arabic spelling uncontroversially reads άλλους and not something else.
The reading is dsta āls. (The first a is an alif, which can stand in for other vowels.) It's not the expected ālws, but I dount that helps: άλλας or άλλες leave us with the same problem in the feminine plural accusative, and άλλης should itself be ālys—or rather, is no more plausible a reading of āls than is άλλους. We're mid-verse, so we can't use rhyme to emend. Was I naive in thinking this is just the indirect object accusative, transferred to άλλους? (I.e. without needing to appeal to the genitive-accusative merger, but purely as an indirect object.) Would this be possible in Pontic? Gotta say, didn't blink about this one at all...
In general, it seems to me that Rumi and Walad's poems present with more problems to the historian of Greek than solutions. :-S
We haven't got much further on this: it's hard to tell whether άλλους is an indirect object genitive, that has merged with the accusative morphlogically in the plural (as happens in Cypriot, and as Belléli thought was happening in the Torah); or whether Cappadocian is already using the accusative for indirect objects (as I'd assumed, and as the Proto-Bulgarian inscriptions were doing four centuries before). You'd need more texts to find out, and you won't get more texts from Cappadocia. (You will get the odd church deed from the Pontus, but there will be very little vernacular there—I doubt there'd be enough to cast light on this issue.)