I would be inclined to tell her that she was an adult when your Father died, and should have had the sense to know that blowing through whatever he left her so quickly was stupid and reckless, and that you are not her personal cash machine, so in future she needs to live within her means in the same way that you have to. Then stick to it.
When my Mum was in her 80's, she said that with her pension, and the various benefits she was entitled to, she had more money than she knew what to do with. So while your Mother may feel had done by, she's obviously not living within her means, and needs to cut back on her meals out, etc.
With regard to your birthday meal out, just hold firm, tell her she wouldn't have been invited if you'd gone away as planned, your DH wants to take you out for the day, and then round it off by spoiling you with a romantic dinner for TWO, and sorry, but no she's not invited on this particular occasion, and neither are the kids, and if she accuses you of lying one more time, she won't be invited to join you in anything ever again! If she makes a fuss, I'd just say 'Mum, I've explained this to you, you're not stupid, so let it go'. Then ignore any further messages.
It doesn't sound like you have anything to feel guilty about OP, so please don't let her continue to make you feel bad. Your Mother sounds selfish in the extreme, and should be counting her blessings, as at her age there are many, many people who are suffering from ill health, which often leads to friends dropping out of the picture, and then leaves them lonely. Whereas she appears to have her health, good friends, and a daughter who has up until now, tried her best for her, but it seems that as far as she's concerned, nothing you do will ever be enough for her. Stick to your guns OP, she may be elderly, but that doesn't entitle her to be rude and demanding.