There are usually two solutions to mounting PCI or PCI-E cards in low-height cases.
1. Conventional mountings but with low profile cards. These still mount the cards vertically, but rely on a card that isn't the full height. These often come with full-height brackets - so they will still work in regular cases, but with an optional (sometimes shipped with the card, often only available on request to the manufacturer) low-profile bracket to replace the full-height bracket.
I have had a couple of full-height card brackets delivered with low-profile cards, and in every case an e-mail to the manufacture has resulted in a low-profile bracket being despatched at no cost. (Hauppauge - kudos to you)
2. Horizontally mounted cards connected to a riser. This can be a rigid or a flexible riser. This allows full height cards to be used, but mounted horizontally, not plugged directly and vertically into the motherboard, but instead into a riser that allows the card to be mounted in a different orientation.
Some case vendors do this to allow full-height cards to be used in low-profile cases, but if you do this with a case not designed to accept the right angled card, you will need to find a suitable mounting system. Also there can be issues with timings, as the riser extends the distance between the card and the PC, which at high bus speeds can be significant.
I'd try and source a replacement bracket for your card first. You may find you can buy a blank half-height bracket and dremel out suitable holes if your manufacturer can't supply a half-height bracket?