Unless you have a very good reason (I compare notes with people at dozens of firms and have never heard one), the only criteria you ever want to get SOC2'd on is Security.

My experience is the opposite of yours: having a security SOC2 ends the vendorsec process it any enterprise buyer, and enterprise buyers virtually never read anything in the SOC2 other than a glance at the exceptions. A very large, very security-intensive vendor we have all heard of told me a story about a vendor they had that gave them several years of repeated Type 1 reports. Went fine.