Hmmm.... I think this is an interesting project... While I don't agree that people shouldn't eat animals, I do agree that we should treat them with respect. For that reason I only buy captive-bred specimens for my marine tank (which I don't eat, but I consider it to be less stressful on the fish and the environment).
If you're going to attempt this, then just be sure you're in it for the long haul... Compared to being improperly cared for and experiencing a slow, agonizing death by the hand of a well-intentioned caretaker; I think being scalded for 1/2 a second in boiling water would be a great way to go... (Heck, I would rather be boiled than ripped apart alive by the numerous marine animals that naturally feed on lobsters!).
Anyways you're probably sick of "advice", so back to your questions... To do this cheaply I would build a shallow tank out of blocks (the large cement blocks used in building construction). Since lobsters are bottom dwellers make the tank shallow and have as much bottom area as possible. Build the tank into a basement floor or something so that you can see the tank from above only. Paint the inside with a waterproof pool paint a white or blue color so you can see the lobster. I wouldn't use sand and a small amount of live rock should suffice if you can use that to seed other base rocks, perhaps just enough to make a few caves in the middle. It would also help to have the tank be circular, since lobsters are known to shoot backwards at alarming speeds and having a circular tank would reduce the impact of ramming into a wall. My thought here is to rely on insulation for keeping the water cool. Find out what the particular temperature and salinity requirements of different lobster species are (I'm sure there are a few varieties available at different stores) and get one that you can reasonably afford to care for. I don't think you should just proclaim that cost is not an issue because eventually it will be... Since they are scavengers I don't think you'll have much trouble feeding them... I think you'll have to be careful though and feel them fish/mollusks or other meaty foods in a feeding trap of sorts so that the mess is contained and you can remove the leftovers when the lobster is finished and it doesn't pollute the water. Filtration should be simple enough, but you will probably need less flow, but plenty of aeration.
Another idea to aid you is to call your local grocers and fish markets and ask whether the lobsters they sell are caught or farmed... While it doesn't help the farmed animals I think consuming farmed animals is generally better than disrupting the delicate natural habitats. Do some research yourself on the costs and quality of each type of lobster and maybe you could convince some of the stores to switch to farmed lobsters. Good luck.