I had my new rescue pony do this exact same thing today - went to try the snaffle on her, she threw her head around, tried to bulldoze past me, reared up, ran backwards, etc.
Like others have suggested, I believe my 19 year old project is afraid of the snaffle because her teeth were terrible and though she just got them floated and they are good now, she doesn't realize it and associates the bit with pain. If your ponies teeth are bad, it will cause a bad reaction.
I am restarting mine with the snaffle though my daughter may only ever ride her in the hackamore - I intend to do a lot of ground driving and flexion and bitting up with mine, I plan on re-starting her exactly as I would a full size horse.
I sent her away from me and made her give me a strong trot for a few minutes, then told her to KNOCK IT OFF - then before I asked again, I tied her head down also - not tightly - just took the lead through her front legs and looped round her girth and tied back on itself - if she kept her head level it was loose - but it did prevent her from throwing her head up - I just made it much easier for her to accept the bit than fight me.
I spent some time rubbing her muzzle and putting my thumb in the corner of the mouth to pry it open - then when she opened it, told her good girl and left her alone for a minute. Then put her headstall up and pried her mouth open with my thumb and very gently put her snaffle on. Finished slipping the headstall on over her halter. Hooked her reins to the halter but left the snaffle on for her lesson - after her lesson went to take the bridle off and she had that snaffle clamped down - so pried her mouth open again and waited until she dropped the snaffle. Gave her lots of praise, let her think about it for a minute, and then put it right back on. When she accepted the bit easily, gave her praise and let her stand quietly with it in for a few minutes, then let my daughter take it back off and with that her lesson was done. We tied her while we cleaned her pen and put her feed in her stall, which caused her to dance around (she's recovering from a body condition of 2 and is still very concerned with getting her dinner) -
If she still would have fought me, I would have sent her out again and made her move her feet until she decided it was easier to just put in the bridle. That and lots of desensitization with the hand on the muzzle and thumb in her mouth to have her open it.
I wish you luck with your pony.
There is no passion to be found playing small - in settling for a life that is less than the one you are capable of living. - Nelson Mandela