Sore gums are a recognised side effect of chemotherapy but I haven’t heard of it as a result of radiotherapy or hormone treatment. (Stupid question - have you read the full list of side effects for tamoxifen?). I’d say you should contact your breast care nurse who will be able to advise you.
The hospital prescribes strong mouthwashes which I found unusable as they stung so much. But a medical herbalist working for Breast Cancer Haven gave me dried marigold petals which you infuse for 10 minutes. Then you use the liquid as a mouthwash several times a day (It’s almost tasteless) and you can use the soggy petals as a poultice if you have a particularly sore area or have an ulcer. It’s certainly worked for me and it’s only about a fiver for a bag from Holland & Barrett or Amazon. No guarantees it will resolve your problem, of course.
All the best for your last two treatments xx