Should I upgrade my electrical panel during a solar installation
Yes I think you should upgrade the panel now, but that is really just an opinion, partly based on disliking split breakers. Making changes after the solar install could get more complicated.

If you decide not to change the panel then no, you shouldn't move the service feeder wires to the directly to the main lugs. You need overcurrent protection for the panel. I do question if the existing stab in "Main Breaker" is Listed for use as Service Equipment, usually some type of screw or bracket is present to secure stab-in breakers to the panel https://www.gexpro.com/usg/Root-Category/Power-Distribution/Circuit-Breakers/Circuit-Breaker---Accessories/Hold-Down-Kits/GE-THQLRK1-Main-Breaker-Retainer%2C-PowerMark-Gold%2C-12-42-Circuit%2C-THQL/p/79046 , I just don't see it. Depending on panel model you may be able to get a kit to mount a main breaker to replace the main lugs. 

$1200 sounds within reason, but that's not a two men for 8 hour price or job.