Sure, you can fix unisons between tunings. All pianists should have experience with this and be able to touchup a tuning.
The hammer will depend on how much you are willing to spend. I’ve used probably a dozen different kinds of hammers in my lifetime. They all feel and respond differently. Consequently, tuners develop strong opinions on what is best. I learned to work with them all. They all get the job done. Ultimately, it depends on how you use your body.
If you are looking for a cheap hammer, amazon looks like they have a $20 version that would work well for a first tuning hammer. “Vktech Professional Extension Piano Tuning Hammer Wrench Lever Hardwood Handle.” I’d stay away from any of the “L-style” or “gooseneck” hammers and the kits. You’d never need a temperament strip anyway.
If you want to spend a few hundred ($200-600), then there are some much better options.
BTW, never use CA glue to “fix” a loose tuning pin. I don’t know who started telling people to do that, but that is crazy talk. It can be used on a junk piano that is already worthless. You can use CA to get some more life out of it. But, it is never how a tuner address a loose pin on a piano that has any value. First the pin gets tapped lower, if that doesn’t work there is a whole series of longer and thicker pins that are used to create uniformity in how the pin response. Once you use CA, you are never going to get it out of the wood without reaming the whole.