This signature weakness acts as more of a deckbuilding restriction: must include cards that heal Ally assets.
It's a very soft weakness if you have the tools to deal with it. It costs no actions, doesn't deal you any net damage, and doesn't ever cycle into your deck. You even gain an extra point of soak with it!
Needing to cure the the Bystander is the wrong mindset. All you need to do is to prevent him from dying. From that perspective things get a lot simpler.
At 0xp Bandages counters 3 instances of damage each and can be scavenged. Soothing Melody counters 2 instances of damage each. In a single deck cycle that packs Hallowed Mirror and Bandages you're already good for 12 instances of damage, which is far less than what most scenarios will throw at you.
With XP, you'd almost always have the Surgical Kit as a backup option, but you should rarely need it. Other very good choices include the Forbidden Tome, Call for Backup, and Manipulate Destiny with blurse support.
Note that with 3-damage burst heals like Bizarre Diagnosis and Surgical Kit, it's more efficient to let the Bystander go to 1 health and then heal back to 4 health than to discard it entirely. That way it can continue to soak damage for you, and you don't waste the heal.