Adding a 100A breaker in front of a feeder
Get a 4-prong outlet and a 4-prong cord, but don't bother with the extra breaker unless you're changing the panel out

Since the box in question is so small (a spa panel, really), I wouldn't bother with the extra circuit breaker if you can find a 40A breaker that replaces the existing 30A breaker in the box.  On the other hand, if you can get the power cut to it safely, then I would consider replacing it with a larger panel with a proper main breaker in it -- an 8-space, 200A, convertible-main mobile home panel with feed-through lugs that has been field-fitted with a 100A main breaker is a good option here, as it can easily be migrated to be a subpanel, or be a main panel instead if you want to convert the existing panel to be a subpanel.

Your choice of 6/3 W/G for the new range circuit is well and good, but you need to use a 4-prong (NEMA 14-50) range receptacle and matching 4-prong cord for your range here -- neutral and ground stay separate at the receptacle as a result, and you also need to pull the bonding strap inside the range when you fit the new cord.