https://support.apple.com/mac-notebooks/repair/service/pricing
"The following prices are for Apple repair service.
Pricing and terms might vary at other service providers."
I think the "diagnostic fee", much like it is at car repair shops, is bogus and simply tacked on to increase the profits of the authorized service center...
but Apple says in the link right above that if you don't go directly through them for the battery replacement that the pricing can be different so it's up to you which way you want to go. Personally I'd go the cheaper route, directly through Apple, to make sure the battery was genuine OEM and the work was done correctly but that's me ¯\_(ツ)_/¯