Nope there's no 3 month extension 😕
Now whether you can or can't get it together document wise to apply from May 4th until your 90 day Non-O stamp runs out May 12 is totally on you and your ability to get the copies, documents, etc that immigrations needs to issue the extension. Did you already go to your immigration office and get the hand out listing the requirements AND ask if there's anything not on the list that you need as well?
As
@ ***************
s correctly pointed out (y) you COULD use the one 60 day visit thai family extension you're allowed per entry and that'd buy you 2 more months and then apply for the year extension after that.
Good luck with it