DSA has the same problem. I don't know what would work well. Subliminal free signatures introduce other issues or are impractical IMO. But smart people are working hard in this field...
Unique signatures are trivial and exist in many forms (e.g. the BLS signature scheme is automatically unique); They all have unacceptable tradeoffs: Huge increases in signature sizes, much less mature cryptographic assumptions, greatly increased verifier computational complexity, patents, one time use, lack of review, etc. (or usually a mixture of all these)
Picking a signature scheme based primarily on your ability to suppress side channels sounds like obsessive micro-optimization. While desirable, we have multisignature already, which is arguably better than sidechannel suppression, and doesn't excessively compromise the many far more concerning criteria.