Hey man, I'm in the opposite situation. I'm a Punjabi Sikh and my girlfriend is Caucasian and Sikh. I'm sorry to hear about your problems and I hope it gets better for you! I'm lucky that my parents and siblings have been accepting, but I do have people in my extended family I know would be against their own kids doing something like this, and here's my advice:

Firstly, you need to understand that for her brother and father, a large part of the worry likely extends from the fact that they feel you will be almost killing their heritage. I've seen this a lot with my friends who date out, but the family generally views the marriage as the end of their culture and religion in the individual's (in this case, your GF's future) family. One of the biggest fears is that your children would turn out to be apathetic towards Sikhi, know almost nothing about their Punjabi side and even if they did, that your grandchildren would likely loose it altogether. I do think this fear (whether justified or not) is actually based in reality, as I have friends who are very mixed (for example, one girl who is partly Arab, partly Bengali and partly British), and the truth is that (according to her) it's pretty much impossible to keep all the cultures alive in situations like this. When her Bengali grandmother married her British grandfather all those decades ago, it also meant that within 1-2 generations, the Bengali culture with her (the grandmother's) descendants would be dead, which is what ended up happening, given the fact that neither my friend, her siblings or her cousins who share that grandmother know how to speak her language or anything about her traditions. There may be exceptions, but based on my own observations and conversations with my friends, this is what I think tends to happen, and for her brother and father, you're going to be the reason Punjabi culture dies in her (your GF's descendants) family.

Your one saving grace is that you say you are interested in Sikhi. If you genuinely are, and wish to raise a Sikh family with this woman, then I think her brother can (potentially) be brought around. Unlike Punjabi culture, Sikhi is about forming a relationship with Waheguru, and if you read Sri Guru Granth Sahib ji, you'll find there's nothing in it which restricts its message to a single culture. Even if Punjabi culture were to end in your family, that would have absolutely no effect on the ability of your descendants to practice Sikhi and fall in love with the Guru, provided you guys do a good job of teaching it to them (and this sub is a great place to learn, I'd definitely stick around!). I mean my parents are very westernised and I wasn't raised to be a traditional Punjabi at all (born and raised in Canada), but that didn't stop me from turning out to be more serious about Sikhi than either of them :p

Sikhi truly is universal and can survive anywhere in the world, and this is what you need to focus on. You need to convince them that Sikhi is more important than Punjabi culture, and that (if you are actually serious about Sikhi) that you have every intention of preserving it and passing it onto your children. This may not be easy as a lot of Punjabi Sikh families are more cultural than actually Sikh, but I think her brother, no matter how against the marriage he may be at first, will come around if he senses that you are serious about your dedication to Sikhi (and not just doing it to impress them) and if you can be the reason your GF also becomes a better Sikh (note: this is different from "doing the right Indian thing"), it will make their respect for you increase even more.

Good luck OP, and I hope it all works out for you. And if you are actually serious about asking questions and learning more about Sikhi, then I think this forum is a great place to do so.