I think some people are hearing OP say "My elderly father needs me to take him to an event, however, I don't want to go. I'm digging my heels in because I didn't receive an invitation so that makes for a handy excuse to tell dad 'too bad'."
What I think she's actually trying to say is "I didn't receive an invitation and I'm okay with that. I'd normally never attend an even without being invited, however, my elderly father has been invited, wants to go, and needs me to drive him. He's really pressuring me to go anyway and he's been trying to get other relatives to invite me. This makes me extremely uncomfortable and I don't know what to do?"
OP, I think you should cut out the middle people and contact the hosts directly. Acknowledge how awkward this is and say something like "Dad has asked me to drive him to the baptism but I'm feeling awkward since I wasn't invited. Please know that I am absolutely okay with that - no hurt feelings whatsoever, but it does complicate getting dad there as he doesn't drive. Do you mind if I tag along or would you prefer to arrange transportation for him?" I think that's your only option other than telling your dad you aren't available to drive him, period.