i can relate mate, to the permission side of things.
my missus and i have pretty good communication with it though. this might be of interest to you; or not but i felt like writing it anyway. It's not meant to be mind shattering-ly inspirational, but it might help get a perspective on it that you might not have otherwise had.
my wife loves baking, and buying clothes and making outfits she can wear out and to work. the 'labour', or her efforts (in cooking and in searching for the right clothes for the right event or whatever) are rewarded by the approval of people when they tell her how amazing her raspberry, blueberry and banana cake is and how nice she looks and consequently how confident she feels when wearing those clothes.
They might not necessarily be hobbies like brewing is, but they require effort and the process involved cost money, the same as brewing does. To make sure she's always able to make nice food and keep up to date with her clothes and shit, she needs to spend money every now and then. This then, keeps her happy; doing what she loves doing.
Brewing is like that for me. Pure and simple. Everyone has something they like to do that costs money - unless you're completely out of luck and she likes walking in the park and nothing more, then i don't know what to tell you haha.
I talked to my wife upfront about it and said, look i want to do this because i have FUN brewing and making the beer. It's something that makes me happy. My reward is that i get to drink the fruits of my labour (even though that part is just as fun as drinking it). Yes i spend a lot more initially to get my pot, my grains, my brewing bags, my storage boxes, my fermenting vessel/s, my bottles etc. Once they're bought, it is very little in return for the amount of beer i make and drink - throwing in the 'after the up front purchases, i can make a case of beer for $x rather than $45-$60 - can come in useful here. After that it's just the different ingredients i buy and the occasional added extra/upgrade i get to my set up.
Having said that, the hardest part was the second fridge but you really have to find one that is cheap - most are though - and for us, the added electricity because i'm a student at the moment so not working full time. You can of course do with less, but it's been a godsend having one.
I'm rambling now and said way more than i probably meant but it's a balance thing i think. If she can understand that you both have interests that cost money, and this one is just one of those things - provided you don't go mental and buy thousands of dollars worth of shit, you should be good. I'd never tell my wife she can't buy that new dress - she's smart enough to know that such a high price tag means we're both eating rice for three weeks straight so yeah, same with you and brewing. you spend some, she spends some, you're good.
space it out too if you can. for me, immediate things i needed for BIAB were my pot, the bags, the thermometer, and the fridge and STC. i didn't go out and buy them all in the one day though.