Is it easier to apply for a Thai visa with a Thai marriage certificate versus a UK marriage certificate?

Well firstly you need a passport, if not a Thai marriage certificate its a bit of a tricky road at the Amphur they will insist your wife is still single, so there are companies in Bangkok that take your passport and certificate get it authorized then take to Intrtnal Affairs and send you back all the authorization for your local Amphur to finally yes you are married and once in their system will issue a letter for immigration when you renew your visa. Think it costed about 4,000 baht to get done but company we used very good.

The yellow book, ID Card or Driving licence is not required for a visa application. If you want to apply for the 90 day Non-O visa based on being married to a Thai you can use your UK marriage certificate to apply for that, however you wouldn't be able to use that to apply for the 12 months extension of stay inside Thailand. For that you need a marriage certificate from your local amphur in Thailand and to get that you will need to get your UK marriage certificate apostilled by the FCDO in Milton Keynes then authenticated by the Thai consulate in the UK and finally you need it verified by the MFA inside Thailand before proceeding to the Amphur.