I run a similar campaign through our channel manager, TravelClick iHotelier. I created a specific rate plan, linked to BAR, and pointed the Google Ads campaign to that specific rate. Alternatively, if it is available, you can create a promo code, include it in the Google ad and track direct bookings using the code. I hope this helps! 

Maybe this is not what you’re looking for, but there is an excellent metasearch platform called WIHP meta i/o where you can run all your own metasearch ad campaigns (Google Hotel Ads, Tripadvisor, etc.), track reservations that come through your ads, and measure your results in great details. It’s very low cost. Highly recommend!

You could try creating a new booking engine, adding a travel agency to the booking engine or as mentioned by Jill a specific rate plan / voucher could be used. 

We created a booking engine specifically for Google Hotels and assigned a travel agency to it, so can track reservations via the Travel Agency. 

You could also potentially track using Google Analytics / Tag manager. 

Like @willjordan mentioned, Mews already have some partners that can handle Google Ads for you. There’s an article about it here.

If you’re looking to manage the ad campaigns yourself then you’d want to set up Google Analytics pixel on your website (you could also do the same for Facebook pixel/Facebook Ads) and connect your booking engine to Google Tag Manager and then to Google Analytics. This way Google Analytics can show you where the traffic came from (ads, organic, referals etc.) what they did on your website and most importantly, if they booked and the value of their reservation. There’s more information on this setup here and here.

Hope this helps!