I would because birthdays can be very depressing. It may not go well but it may just be what he needs. You can ask him what he thinks but this may be another chance for him to draw into himself or to isolate.
I think you should be worried to a point. He is under a lot of pressure but he also the kind of person that is getting a PHD (highly functional) so support him in any way possible. Sometimes people are too proud to ask for help
I think your heart is in the right place. I am just concerned that it may not be good timing. On the other hand, it may be just what he needs. Try fo find out from close to him that knows his educational load right now and see if he can use the pick me up