Excellent !!! I was in a somewhat similar position a few years ago, except in my case I didn’t have a job. I had two job offers offering similar wages. I negotiated for a higher wage with my preferred option. When I got some push back, I played the “I already have another offer” card.
My family thought I was crazy to bargain for more. Alas, the offer was raised by 10% and I accepted.
I think women are generally expected to be more cautious and less aggressive with things like that. While I’m all for caution, you’ll never get that extra if you don’t try albeit reasonably.