The part with 'tell her you are just playing play money' sounds like a very bad idea to me. I am a woman myself and for sure she will found out you were lying 🙈.
In my case My boyfriend started playing poker. First I was very sceptical myself, but after he explained me how it worked I was convinced he was playing with an innocent amount of money and I even started playing myself.
So maybe it will work to watch some livestreams on twitch together, learn her some strategies, open an account for her and start with some freeroll tournaments. At least you can try? For sure I would be very very mad if I would find out my boyfriend is lying about how much money he spend.
Is "His money His money", or Is "His money your money"? For you to get mad about "His money", as If It were "Him spending your money, Is absurd". While, everyone likes to "Spend money, and buy things, and Just have money to (Have money)", It's another thing to (Control), someone else's finances.
No-One has the right to, "get upset over someone else's spending habits", Especially If It's not "Your money that the said person Is spending In the first place".
Now, had you both, decided (We're gonna have a Joint bank account), and each paycheck, we're gonna save X/Amount from our pay cheques In our (Joint account), and split the bills 50/50, and the have a couples night out, and keep X/Amount to do with (As we each well plz separately from our own pay cheques). Than you later find out, that (Your partner has been dipping Into the Joint account), and spending/splurging on It on poker (That's understandable), for you to get upset over.