Hi, It is your TDEE not your BMR that is relevant for this. You’ll find that your TDEE is much higher than your BMR, although it’s likely to be less than the usual example of 2000calories. For most older women our TDEE can be an unpleasant surprise.
As to whether you need to stick to 1/4 of your TDEE on fast days – that’s up to you. Michael Moseley has said that both 500 or even 800 are OK. It depends on what you are trying to achieve. If you want the health benefits of fasting without much weight loss then 500-800cals would be fine. But obviously the fewer calories you eat the quicker you lose weight, so if that is your goal then I’d aim for somewhere between 1/4 of your TDEE and 500cals.
What is important is that you choose a calorie level you can sustain over time. There’s no point choosing 1/4 of your TDEE and then giving up after a week because it’s too hard. Start at a level you feel you can cope with and then adjust down if you are able.

My TDEE is just 1400cals, so I really should be having 350cals on fast days. I started at 500 and over time found I could manage with less. I now usually have 350-400 on a fast day, but I only cut out some fast day foods as it felt comfortable to do so. Over time you lose that fear of being hungry and it all becomes much easier.

There is nothing magical about 1/4 TDEE. This is a concept that was plucked out of thin air by Michael Moseley as he thought most could achieve this. I for example eat zero calories on my fast days. Others eat 500 or 800 or 1000. Work out roughly what your TDEE is and reduce it by as much as you think you can handle.