That's a tough one. I'd say there was a failure on your part for not asking what it'd cost to have them send a tech out initially. I'd also say they're likely padding the bill because you said you'd order the parts yourself instead of having them supply parts and make money on the parts markup in addition to labor charges. 

From your perspective they billed you $400 and didn't even give you a quote for the work. From their perspective they're dealing with a customer who wants to save money by sourcing his own parts and only wants to pay labor. 

I think you'd be best off to send the $200 and be done with this company. No one is going to be happy with that resolution but consider it a lesson learned and do your own work moving forward or at a minimum ask for pricing upfront.