If you spend a little while reading threads here on this board you'll come across plenty of mums who are endeavouring to get adequate child support for their children from fathers who don't wish to pay, and also mothers who are trying to protect their children from abusive fathers but are legally obliged to ensure regular contact.
It's a farce.
What more do you expect your gf to do? If her ex is committing fraud what is she supposed to do about it? Bear in mind her actions will be driven by whatever is best for her child, regardless of her (or your) personal feelings.
If you're moving in with her presumably you're in for the long haul and foresee a committed future? If so, then you will for all intents and purposes be a second father to her sons simply by nature of being the 2nd parent and authority figure in the household and by contributing financially to household and family expenses.
When joining a family to create an existing family, you need to be sensitive to existing relationships. The best advice I can give you is to step up and treat her sons in the same way you would wish your child be treated should the situation be reversed.
Ultimately, there's not much, if anything you can do about this ex who doesn't bother with his son. You either accept this child as part of your family or you don't. I'm not saying it's easy, blended families are difficult to get right for a reason, but remember the child is innocent in this.
Your own child isn't likely to miss out in anything materially significant in this scenario and you will all benefit from a harmonious family set up however finances are divided.
Of course you need to assess finances and no one is judging you fir that but be sure you don't alienate this boy. It'll be a huge adjustment for both of them having you and a new sibling in their lives.
If the ex cannot be made through the csa to contribute properly then really all it cones down to is can you afford your new family and are you prepared to accept them as whole?
Don't commit before you're sure because it's not just your heart you're risking- there are 3 innocent children in this too.
Take it slow.