No, there’s no ‘nice’ way to get the pistons out without a compressor. If you need to change the piston seals then my advice is to change the pistons too. In this case you won’t need a compressor. Before you take anything apart pump the pistons out all the way. If one is lazy, hold back the other one so they both come right out. One will eventually pop all the way out (making a mess). This is fine as long as the other one is sticking right out of the calliper.
Then you can disassemble the calliper and just grab hold of the remaining piston with molegrips / pliers / whatever and drag it out the rest of the way. I think I ended up levering one out with a screwdriver, mangling it in the process.
The point is that if you don’t need to re-use the pistons you can remove them any way you like (as long as the calliper body isn’t damaged) and £7 for a pair of pistons is cheaper than buying a compressor.
You’ll end up with dot fluid all over the place – I worked in a big plastic tray. Have a spray bottle of water handy to soak the bits and bobs that get DOT fluid on them, or it’ll strip all the paint.