Yes, sequencing of overlapping regions is a good workaround in this case.

Since you are in the US cleanup kits and primers ( desalted are ok for pcr and sequencing primers) I would use 3 primer sets with and overlap of 100 bases . The shorter pcrs will amplify better and easier as secondary structure will be less of a problem but it will cost a bit more purifying 3 amplimers and the 3 shortmers will probably take more time to get the pcr conditions right so there is no real problem with a single template and sequencing primers that overlap by at least 90 bases. Both methods work well so I suppose it depends what you will do next with your results. Possibly if you are looking for changes in many samples in the future it might be easier to have short amplimers and a slightly easier pcr protocol

2500bp is totally reasonable size to amplify in one PCR reaction. And you can clone it into a plasmid* too. Easiest way to have all the DNA you'd ever need.

You can typically get 1000 good bp per Sanger read, but not all primers work well for DNA sequencing. So, plan on trying out a few.

If you get one bright band, you use use ExoSAP-it as a cleanup without losing any of the product. 10 microliters PCR + 1.0 microliter ExoSAP-it reagent; incubate 20 min 37*C & 20 min 80*C to inactivate the enzyme. Add in your primer and it's good to go!

*cloning into a primer means you can use "universal primers" that flank the cloning site. Those work very well and should get you most of the sequence data you need.

Good luck!

Have you considered using an outside company like Plasmidsaurus? They use Oxford Nanopore’s MinION platform, allowing full-length sequencing of the 2.5 kb amplicon in a single read. This eliminates the need for primer walking or tiling strategies required with Sanger sequencing.