OP is travelling in a group of 2-3, and twin/triple occupancy does not necessarily mean x2/x3 room rate. This would expand the options.

Gion Fukuizumi is what I mean by 'standard'. (People may call it 'budget+' ) Starting at JPY 8400 with breakfast.

Yes, Three Sisters Inn Annex is Historic, rather authentic, yet Gaijin-only Ryokan in the OP's price range. Located within a few blocks from Heian Shrine, and is often recommended for ryokan beginner.
FYI, you canNOT book on the web or by email.

And Tomiya is 'budget' at JPY8400+ (with dinner/breakfast).