Hi, there are a lot of guides that could tackle that.
Quite a bit of them focus on their local area, so depends a bit on where you are going to climb.
In October you might get lucky and have conditions for water ice, but it’s more likely that you would climb on hard ice, in one of the outlet glaciers near the south coast. Sólheimajökull is a part of Mýrdalsjökull and is about 2 hours drive from Reykjavik, would probably be most likely.
I would personally contact https://www.mountainguides.is/, a rather large company by Icelandic standards, and good quality guides. They have operations in Sólheimajökull whole year round. Both the companies that you mention have high standards, but one focuses on Heli skiing in the north and I think the other is waiting for Covid travel restrictions to ease before resuming business.
https://www.mountainguides.is/category/glacier-tours-solheimajokull
Have fun,
Sissi