First of all, if the dealer winterized it like you said, they should take full responsibility for damage as a result of them not doing the job right. They have no excuse since they had it all winter. Now, if the problem is something unrelated to the winterization, like the heating element, that's a different story, even though they may have accidentally hit the switch or something. But if the problem is a cracked tank or line, they should take responsibility.
If the problem is something like a heating element, they should have tested this before they sold it to you unless you bought it as is. If I were running an RV dealer, I would want to check all appliances on a unit before winterization to make sure everything works and I'd know if something would be my fault or not.
I would take this issue farther into the company, possibly the service manager or store manager. Customer service is (well, should be) their # 1 priority, and if they care at all about you, they should get that heater replaced free of charge. If they don't do anything, I wouldn't bother going back there. Surely they can do better on the price, because that isn't even 10% off.
I would start my asking them how much you mean to them and if you mean anything, tell them to give you the heater at cost to them, plus the labor.
Another thing is I really doubt you need a whole new heater. They sell replacement tanks and heating units separately, which will save a lot of money. But, I don't know all the details, so I can't make a conclusion.
Now about installing a new one...
It should be fairly easy to do assuming you can get the back part of the bed platform off, which you probably can by removing the hinge screws for the movable part and the screws holding it down. They are probably all square head. Once that's off, you can get your hands down in there to disconnect the gas, electric, and water connections. Make sure all those are off before attempting anything. Make sure to have windows open because the line will probably have gas in it that will escape once you open it. Next, pull out all the screws on the outside flange and the unit should pull right out. In order to install the new one, make sure to have some new putty tape to wrap it with. You don't want to use the old stuff. Then, simply reverse the process, making sure connections are tight. Check the gas connections with soapy water and then put water to the system and check the water lines and tank for leaks.
I believe I heard somewhere that it's easier to slide the tank out like 3-4 inches and then disconnect the lines. It gives you more room that way.
Also, if the trailer is still winterized when you start the process, and since it was probably winterized with antifreeze, make sure all antifreeze is drained or you'll have a pink mess everywhere when you disconnect fittings, assuming you don't have bypass valves.
2007 T-286SR Cherry/Granola, #6236, original owner, current mileage: 9467.8 (as of 5/26/19)
1997 T-2653 Blue Denim, #5471
1979 12 1/2' MC, Beige & Avocado, #4639
Past Sunlines: '97 T-2653 #5089, '94 T-2251, '86 T-1550, '94 T-2363, '98 T-270SR