I would buy a syringe of easy silver solder paste from a jewellery supplier such as cooksongold.com. Unlike the poor service of another well known jewellery supplier criticised on here recently; their customer service and delivery times have always been spo-on when I have ordered from them.
Jewellery silver solder paste is much 'softer' than Fryolux type soft-solder paint and very controllable in application (its primarily used to repair very fine jewellery where there is a risk of either melting the item or there oo difficult to control 'normal' silver solder pallions
It can have porosity issues so whilst I wouldn't recommend it for 'structural' situations like caulking, it should be perfect for your rivets