Looks like a cool watch, and super cool that it comes with a Jubilee-style bracelet! Your search for a mechanical movement will be physically restricted to the movement dimensions (diameter, height) in order to fit in the case of the watch you currently have, and I haven't seen too many salmon colored dials with a day-date movement for cheap. But if you have dimensions for the inner diameter of the case, and whether the dial comes through the front of the watch (crystal side) or through the back of the watch (case back side), then those details will also be important. Might be a lot of work to take on though.
If you like this look and don't mind getting something else instead, it will be easier to buy an existing mechanical watch that has a lot of aftermarket support for it, say for example the Orient Kamasu. There are also some popular modded Seikos (search Seiko 'Datejust') that look pretty similar to this,
I always wore my Luminox blackout, which is essentially a lightweight plastic case on a rubber strap. When I eventually decided that I want to get another dive watch to replace it, I got a watch with a steel case on a rubber strap, and I hated wearing my new watch. I hated wearing it because it was top-heavy and didn't allow my wrist to breathe, so it was a clunky mess to wear. I love wearing my Christopher Ward on a bracelet nowadays, but sometimes a heavy watch feels suffocating. Maybe I'm just used to lighter watches.
To me, I think it really depends on your father. If you think he'd be offended at the idea of exchanging the watch for something else, then don't bring it up and at best it sits in a little box that you can see occasionally and remember. If you think he'd be receptive to the idea, then at best you have another great sentimental piece. This decision shouldn't be as fatalistic as other people are making it, in my opinion. The watch doesn't represent your guys' relationship, you and your dad do.