The 308 is a great cartridge, but if I was looking for new rifle that is made to handle a 4-500yard shot on a big bull, It would be a 7, or 300 mag. Of course many other lesser cartridges can do it when the stars align. The big thing is "You" need to have faith in the rifle combo. When the chips are down, I grab my 300 Win. I also hunt with 6.5x55, and 7-08, both will do the job, but the 300 is so far my choice when it just has to get done
I also agree your 88 isn't up to the task at 4-500 yards. Good on you for recognizing that now, don't wait for September.
PS if I didn't already have my 300 Win, I'd probably go 7Rem.