Hey Don how are the cats doing ?
I cannot added anything regarding the components +/- battery that has not already been said. BUT
Is the board conformally coated? I ask cause this can be a fly in the ointment when de-soldering. As said those holes are through plated, so as you now gentle is way. I assume from what you said you have the vacuum de-soldering bit of kit. The one tip I can offer is when de-soldering a large IC like shown is to place a dab of heat sin compound on top. Then put an appropriate sized piece of Al on top of it, just decreases the chance of cooking the chip. I usually put a socket in so do not have to worry about too much heat going into replacement & if you need to remove it again for testing or what have you, much better. Bearing in mind that some ICs are not socketed for a reason, longer the pin to board distance the more apt it is to possibly pic up interference.
Good luck, hope my little anecdotes, are of some help. Solder wick is great but I use to clean the holes after extraction, using it in place of a proper sucker IMHO requires too much time to heat, thus more heat into IC, it great for quick removal of most other discrete components resistors caps etc.
Regards, Doug