How can I measure the impact of brand marketing campaigns on my product?

I think there are two approaches to this, and the one you use depends on how you think about your brand's contribution to your reach.

If you believe that the majority of customers hear about your product through paid campaigns, then you want to weaponize your brand to make your performance marketing more effective. I think the best I've ever seen this approach explained is in this interview with Gabe Leydon, the former CEO of MZ:

https://www.youtube.com/watch?v=oXBqzpExvrk

The idea here is that your brand activities are just a means of improving the performance of your paid campaigns: brand recognition makes people more likely to click on your ads. If you take this approach, then your brand spend needs to be equal to or less than the dollar-value of campaign performance improvements it is driving. To set this up as an example, let's say that you're spending $10MM in brand advertising media each year and $100MM on direct response media. To pay for itself, the brand advertising spend would need to generate an improvement of 10% on performance marketing campaigns, either through a decrease in acquisition costs or an increase in acquired-user LTV.

Taking the other approach -- you believe that most of your reach is attributable to brand recognition / brand equity -- then you should be focusing your measurement energy on understanding the incremental impact of your marketing spend (eg. what your marketing is delivering that you wouldn't have gotten anyway because of your brand reach). Incrementality is an active research field within marketing science at the moment and there are a lot of different strategies that people use to determine whether campaigns produce incremental value or not (one such strategy is the Ghost Ads model).

With this second approach, you'd try to find the optimal level of spend that produced verifiably / reliably incremental revenue; with a large enough brand, at some point you reach diminishing returns on ad spend, so this second approach involves iterating through spend amounts and targeting configurations to discover what that level of spend is.