You've probably already read Peter Ellis Jones's splendid book 'Bangor 1883-1983: A study in Municipal Government' which provides much biographical detail of Bangor councillors. Unfortunately he doesn't give Richard Jones's political party or ward.
If you know the date (at least month and year) of any elections, you could look at the North Wales Chronicle. Was there an obituary in the Chronicle? You could also look at the Council Minutes - they're in printed volumes - at Caernarfon Record Office.
I wonder if he was a member of the Bangor COPEC branch? They were passionate campaigners for housing reforms and built the houses in Seiriol Road (1926) to prove to the Council that it was possible to build decent working-class housing. The houses were designed by Arts & Crafts architect H.L. North. Members of COPEC (Christian Organisation in Politics, Economics and Citizenship) were from various chapels and churches, and were advocates of the 'Social Gospel'. There's some info on this website- go to Tools & Features - Historical Sources - Building Schemes. Was Richard Jones active in his church or chapel?
In order to answer your original question, I still think you need to check the council minutes (Housing Committee) and possibly the Chronicle (especially 1935/36). If you live outside the area, it may be more cost-effective to pay for a couple of hours research by Gwynedd Archives & Museums Service.