I'm tempted to try a set of 16s on it. Can anyone see any problem with that appart from it looking like a roller skate?
I have a GT 1.9 JTDM, not a Cloverleaf but I don't think the Cloverleaf suspension makes a big difference because all GTs ride the way you describe as standard. Like you I found the ride shocking and I didn't want to spend £000s on it. I also came to the GT from a 3-series (not an M3 though!).
Mine was on 17s, it's still a bad ride on 17s, so first I tried 16s. This did make a big difference. The car doesn't look as good if you were to compare the two alongside but it doesn't look bad either - the 16s are multispoke and an attractive wheel in themselves. Remember too that Alfa sold some GTs with 15s! The ride is worse in winter (colder rubber, plus more potholes) so for one year I used 17s in summer, 16s in winter. Not a bad compromise, and cheap.
When I got a bit of money available I decided to upgrade to Koni FSDs. Unlike others on this forum I kept standard springs to save money. I was convinced the problem was weak damping rather than soft springs. This proved to be right IMHO and the car is much better. Only problem (apart from £900 cost) is it took the car a while to settle on the new dampers and the front especially looked a bit high after the switch. However I have met this problem with new dampers on other cars previously, I knew it would settle. The ride now is supple and the handling just as good as before, probably better because less likely to be thrown off course by bumps on bends.
The ride is so much better with FSDs even on 17s that I have decided to keep with 17s all year round and sell the 16s. They are still the best cheap fix (so PM me if you're interested, approx £200 the lot inc. 4 decent tyres, one new, all MOT-capable!).