Airbnb seems to have several suitable offerings. For example, this 'self-contained studio garden flat' is priced at £485 for 2 people, with pets welcome (using Sat 12/09 to Sat 19/09 as sample dates):
https://www.airbnb.co.uk/rooms/3628075?adults=2&check_in=2020-09-12&check_out=2020-09-19&source_impression_id=p3_1580323470_bZl3LRfy1m8b0dU7
Floirence Cottage is priced at £484 for the same period, and described thus: "Our place is good for couples, families (with kids), and furry friends (pets)":
https://www.airbnb.co.uk/rooms/13131587?adults=2&check_in=2020-09-12&check_out=2020-09-19&source_impression_id=p3_1580324053_dB1F7v8Yn5umEXg%2F
I'm hoping that those links aren't the sort that time-out after a certain period but you should still be able to find the properties fairly easily on the Airbnb website anyway. Start here:
https://www.airbnb.co.uk/s/Exmouth--United-Kingdom
Enter you're preferred dates, change the number of adults to 2 and carry out an initial search.
Click 'More filters' and select 'Pets allowed' (under 'House Rules').
Take an initial look but then, if you think it might help, click on 'More filters' again, click on 'All property types' and then on 'Cottage'. (Also try 'Chalet' though, as that brings up quite a few possibilities too).