As mentioned, it is a fair distance between Palenque and Yaxchilan / Bonampak. To make the excursion down there worthwhile, I would think you would want two days. I suppose that technically you could drive/transfer to Frontera Corozal the afternoon/evening before, go to Yaxchilan in the morning, Bonampak in the afternoon, and be back in Palenque in the evening. If your goal is to absolutely see the two sites but only have one day, that would be the way to do it - but poor for birding and not really taking advantage of the area.
I would recommend spending the night in Frontera Corozal and going to Yaxchilan in the early morning. There is a decent and fairly priced hotel with a good restaurant near the docks called Hotel Nueva Alianza. It works well to visit Palenque until early-mid afternoon, then get a transfer to Frontera Corozal, arriving early evening. When you arrive, you can head down to the docks and arrange a boatman to take you to Yaxchilan at dawn. Getting there early is lovely as the dawn ride on the misty river and having the ruins to yourself for an hour or three is fairly magical. It's actually one of my favorite ruins anywhere, and the morning ride on the river, the atmosphere, and being there alone are a large part of what makes it so amazing. I don't recall what I've paid for this in the past but it was reasonable, I want to say something like $80 US for the private boat rings a bell? In any case, like I said, I'd really recommend arriving the afternoon/evening prior and arranging a private boat so you can go at dawn. Birding at Yaxchilan is downright excellent, and the main clearing is (at least seasonally) one of the best places to see Lovely Cotinga so it's worth spending time scanning the canopies there. I've never missed them in three visits. You will be back in Frontera Corozal sometime late-morning to mid-day. There isn't much birding to mention around the town of Frontera Corozal, but there is around the town at Bonampak, so I would transfer to Bonampak for the afternoon. Personally I would stay at Campamento Rio Lacanja and bird their trails and that end of town (up against forest) and/or go to the waterfall trail (I think it's called Sak Nok) for good afternoon birding. There are a few options to stay in Bonampak - as mentioned I've liked Campamento Rio Lacanja for good birding out the front door, and the food is also good. To visit the ruins of Bonampak themselves, private vehicles are not allowed to enter. The locals will take you to and from the ruins for something like $20/person, don't recall the exact amount, and you can probably bargain a touch. However, if you have a full day to visit, the entrance road is tremendous birding and can be walked and birded. It's like 5-8km, don't recall exactly. I've done it a few times. I usually will have a couple of folks stop and ask me what I'm doing and/or pressure me to take a guide or to pay for a ride. I've been told that it's dangerous and I will get lost, which is patently ridiculous, it is a flat straight well maintained dirt road with no forks that leads directly to the ruins. It is understandable that they want to protect their business of transporting people and I don't hold that against the community at all, I just personally prefer the early morning walk to the ruins without a "guide." I've been successful in explaining myself and being allowed to keep birding. Then I happily pay for the ride back to town after, just catching a driver at the ruins (sometimes with a bit of delay). The ruins themselves are not terribly large but are very interesting, you could see them in 1-2 hours if you were in a hurry or could spend a few hours, particularly considering that scanning from the ruins is good for raptors (King Vulture is quite reliable).
As others have mentioned, Palenque is good birding as well - fewer "special" birds but very birdy and diverse all the same. The ruins are magnificent and merit a lot of time if you're really into ruins so don't get too distracted by the birds and miss out on the site itself! Palenque is chock-a-block full of people and vendors in comparison to the very laid back / almost empty Bonampak and Yaxchilan, just to mention.
As far as how to get around - I am guessing you'll have to contract a driver or book transfers, no idea on costs. I've always had a car when there. If you do end up with a rental car, two things to be aware of. First, in MX you will almost always be forced to take additional insurance that won't be reflected in your initial internet booking - it's very annoying and will add something like $10-20/day (unless, perhaps, you book through a smaller local agency). Second, if you are out and about driving around on your own Chiapas is fairly safe during the day but you should not drive at night and you might end up stuck in a roadblock at some point (particularly if you take the Ocosingo Rd from Palenque to San Cristobol).
Good luck & cheers,
Josh
