Katie...
I know this must hurt!!! BUT it sounds like your daughter is lovely, intelligent and well mannered. She will work out (and probably already has) what is going on here.
I am one half of a seperated couple, and my children quickly worked out their dad was treating me badly, even though I never said a word about him.
I think the idea of maybe suggesting another word for mum... might be better? Maybe even MumTash, or something like that so she can 'please' her dad and still not feel disloyal to you.
I have much respect for the way you are handling this. It appears your daughter already feels uncomfortable with what her dad is asking her and will not suddenly get more comfortable with it as she gets older. My mum remarried when I was 8. She insisted we called our stepfather 'dad' even though he was an evil person.
We did this in public, in private (only to each other!) we called him by his name, soon we dropped the dad and just didnt' refer to him at all.
You love your daughter, she loves you, she will NEVER get confused as to who her mum is.
Poppy xxxxx