Happy Early Birthday!
I would pick about four restaurants that you might like and call back and try to get a reservation for your birthday. You might have better luck than you think with snagging a reservation. I've been booking and canceling ADRs for our trip in 21 days and having no trouble doing so, either!
If that doesn't work and you still aren't able to secure a TS meal for your birthday at a decent time, I'd suggest trying to have concierge services at your resort book one for you (assuming you are staying on property).
Another thing to consider, you should be able to get an ADR at one of the resorts. Some of the more popular restaurants in the parks might be booked, but I have found that table service restaurants in the resorts can still be very accessible and flexible with the times available.
Here's where I'd recommend trying to dine for your birthday!
1) California Grill -- In my opinion, it just doesn't get better than this! If you are going to be in the Magic Kingdom earlier in the day, you can just hop the monorail and you're there!
2) Narcoosee's in the Grand Floridian (again off the monorail).
3) Citrico's in the Grand Floridian (same reason as above).
4) Kona Cafe in the Polynesian (same reason as above).
I'd suggest 'Ohana but that reservation would most likely be as difficult to get as an ADR at Le Cellier.
5) Wolfgang Puck's Dining Room (Upstairs) in Down Town Disney
6) The Brown Derby in Hollywood Studios.
If you HAVE to wind up doing a counter service for dinner, I'd make it somewhere other than a theme park. You don't want a theme park burger for your birthday dinner!
I'd get creative and go to Beaches N Cream and get a great burger and then have some insane sundae for my birthday dessert in place of cake! Or, I'd go to the ESPN Club or Wolfgang Puck's Express.
Good luck and Happy B-Day!