I use Swissstop pads or brake blocks on all my bikes, and think it might be worth trying them over new callipers. However, if you buy new brakes they might come with cartridge holders, in which case you’d only need the Swissstop brake blocks.
Another option would be to just upgrade the front calliper – I rarely use the rear brake, but I do rely on having decent stopping power on the front. Also, you don’t have to use the latest Shimano callipers – anything made in the past few years should be fine.
I splashed out on a single Dura-Ace calliper on a fixie about 10 years ago, and never regretted it – it worked superbly with Swissstop pads, and that was on a bike with only a front brake (plus the fixie rear ‘brake’). I’m in Scotland, so it saw a number of super-steep descents in wet conditions, and I never felt under-braked, which was more than could be said for the standard callipers and pads.