I don't know about Braintree, but I can report that for my SaaS app (https://www.wisecashhq.com, currently priced in $ but France-based), I used Recurly (https://www.recurly.com) which handles EU VAT very nicely (including VAT changes happening in 2015).
I use them together with Stripe as my gateway, it generates proper invoices and applies the rate of the buyer's country unless a valid VAT number has been provided. They also provide a stripe-like js library so that the credit-card number doesn't go through my server.
With regard to VAT MOSS report, they do most of the checks, I only wrote a little ETL script using http://www.kiba-etl.org/ to create a break-down of the related amounts for each quarter and each country, which I then submit manually to the VAT MOSS French website.
So in my case, this is fully automated, and I'm pretty happy with this :-)
I use them together with Stripe as my gateway, it generates proper invoices and applies the rate of the buyer's country unless a valid VAT number has been provided. They also provide a stripe-like js library so that the credit-card number doesn't go through my server.
With regard to VAT MOSS report, they do most of the checks, I only wrote a little ETL script using http://www.kiba-etl.org/ to create a break-down of the related amounts for each quarter and each country, which I then submit manually to the VAT MOSS French website.
So in my case, this is fully automated, and I'm pretty happy with this :-)
Feel free to email me if you have questions.