I’d leave the existing finish on, fill the dips, then sand and add more finish over the top. I use standard fine texture wood filler for jobs like that, though there are other options like superglue if you want.

Most of the cheaper Fender-style guitars (including Squiers) are fitted with mini-pots and the cavities routed accordingly. The Tele cavity is small enough at the best of times, and hasn’t got a lot of space for additional wiring if you do more complicated switching options.

If you’ve got a Dremel or similar, then one of the small sanding drum attachments is good for enlarging cavities, and is generally safer than drilling. Put some masking tape around the cavity, mark the control plate outline and be sure to stay within that when widening the cavity.