Need some advice on dealing with Evans

Can you ladies and gents confirm that there's definitely a compatibility issue here?

Yes

-To what extent could wear in the chain affect how well it fits onto a cog? If my chain was brand-spanking new, would I at least be able to get it to work?

No, 3/32 is narrower than 1/8th and will never work. A worn chain of the right size might not work well on a new cog (and is not advised) but would certainly fit over the teeth.

Am I within my rights to ask for my money back?

Pass

What's the best course of action?

Buy a new 1/8th chain, fit to fixed side, live the dream.