There are a million and one rational reasons why she’s not attending the sleepover before you even take any argument into consideration.
1 - Her mum doesn’t know you or your daughter. Her own daughter has gone “can I stay round X’s house tonight?” and her mums like “eh? Who’s that? I don’t even know them. No you can’t”. She probably doesn’t even envision you’ve spent money and time buying snacks or planning activities because she doesn’t know who you are and thinks this is probably a random last min sleepover thing requiring no effort.
2 - The girl didn’t want to go to the sleepover so made it out like her mum wouldn’t let her (completely normal thing for teens to do as makes them out not to be the bad guy)” or has told mum she doesn’t want to go but doesn’t know what to say as an excuse so mum has said “just tell her I wouldn’t let you go”.
3 - It might have not been last minute and she could have been banned from the sleepover much earlier but hoping her mum would relent so left it to the absolute last minute to let your daughter know she couldn’t come.
4 - Family might already have had or want to have bank holiday plans. They don’t even know you or have your contact details.
5 - The argument or incident leading to the ban could have been substantial enough to be grounded. It could be anything. Bullying, crime, drugs, theft of money, police involved, breaking curfew, lies, violence, repeated insolence or vile behaviour, refusal to to simple chores or tasks conditional of being allowed to attend, calling her mum a c*t. Could be anything! You’re assuming the argument was over something minor so you think this is disproportionate reaction by the other mum. You have no idea. The other mother clearly felt the appropriate punishment was not to be allowed to attend the sleepover. Thats her parenting right.
You might think that there’s no behaviour serious enough to warrant a ban from a sleepover but other parents don’t feel this way. There’s no right or wrong here. They are parenting as they see fit.
We’re not talking here of inconveniencing or hurting people in a significant way such as if the two girls had concert tickets or had arranged a trip together where one person not going would significantly effect the other person or cost them a lot of money. A low-stakes scenario like an attending a sleepover, going to the cinema or meeting your mates in the park to hang out are all perfectly normal activities to be grounded from without a second thought. These are completely non-events. Your daughter will have to (gently) just get over it.