Cake Height One Baked - Do I really need a springform pan?

Should work with the deep pan that you have. For some insurance, create a tall collar with some parchment, in case the batter rises past the top of the pan.

I think that should be fine! The cake will rise a bit while baking, so if you find the batter is getting very near the top of the pan while pouring it in, just use a bit less—you can bake off the excess as cupcakes in a muffin tin!

How deep is your cake pan? I have two springform pans and they’re both just under 3 inches tall. If your cake pan is not that tall, you could reduce the batter proportionately (I.e., by 1/3 if yours is 2 inches tall). Or you could plan to make it in two cake pans (if you have two) and just have two thinnish layers with maybe some jam sandwiched between the two. A Bundt cake pan would probably work, too. Just make sure you butter and flour it really well. Hope this helps.