No, I think you would find you need a new LNB, and that would take some modification, a bigger dish will help but its not the answer to tree growth, the only thing that will help is a clear line of sight, so the tree will have to go or the dish will need to be relocated. You can purchase a 30" dish and DBS LNB and it will help, but you need a clear LOS.
I can attach the LNB from my 18" dish easy enough and give it a try unless you think its an exercise in futility. Simply put, when it rains and the leaves get wet, I loose almost all of my signal. Once the leaves dry, my signal is back to normal. I am hoping that the extra gain from the larger reflector would do the trick to amplify the little signal that does punch through.
It will help in the short term, might get you by until winter but by next spring you will be in the same position, its worth a try you can always go back to the smaller dish, I guess you have tried repeaking the small dish before you start messing about with the bigger dish, almost every dish I have come across could be fine tuned slightly as over time they move, even the best install, the roof moves from season to season, the ground moves....
If the hole for a 18" dish gets blocked by wet leaves why should a 36" dish be able to get signal? Yes it will work for reception with a Directv LNB u bolted to the mounting point. I am using an old Directway and an old Primestar dish on my home to add gain for rainfade. But you have to get signal to the DISH and your description is loss of signal getting to the dish due to obstructions(leaves).