You can definitely remove the axle if the limbs are under stress in a press. I do it all the time. I not sure about your bow, but if it has bearings in the cam/idler then you don't need to lube them. If it's bushings then by all means give it a try. I suggest using white lithium grease on the axles/bushings.

Just pay attention to how any washers are located when you pull things apart. You may have to tap the axles out. Just don't get carried away. A twelve pound sledge isn't necessary, LOL.

Some guys may say just take it to a shop, and yadda yadda yadda. You sound like you have a general idea of what to do, so how are you going to learn? I always say it's good to know and do this stuff. Then you have better ideas of what's going on if the bow starts shooting goofy.

To explain more, when I get a new bow I mount a rest and set the drawlength and chrono the bow. This gives me a general idea of what performance I can expect. Next step is complete disassembly and throw it in a pile on the floor. Inspect every little part and then put it all together again.
Goffed up a few times, but figured it out. Then got smarter and started drawing pics of how the rigging fit the cam. Now all I do is take a digital pic for reference.

Go for it.