What are you planning on trimming? Taking off some lower branches on your side won't hurt the tree if done properly...it will only annoy your neighbor. However, I would let him know what you are going to do before you do it, even if he won't like it. And I suggest that you model good neighbor behavior by telling him you know your back yard needs to be cleaned up, and then do it.
If the base of the tree is very close to the fence line, I would probably build up to it from either side and leave a little clearance. You can also choose to have the fence jog around it, but I wouldn't recommend that because that will result in your neighbor being able to legally claim that little area as his own some years down the road.
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You really need to weigh your time and effort with savings. Shopping used furniture can be an all-consuming task with day after day trying to find specific pieces. I have yet to have found a second-hand store that offered everything I needed. Robb & Stuckey has an outlet store in Fort Myers which is well worth the drive. The prices are 70-80% off retail. You still may pay a little more than a thrift shop, but sometimes the convenience of buying everything in an afternoon is worth it. In additon, they offer white glove service any will deliver and install everything you buy--work on getting them to include this for no extra fee. Everything in the furniture business is negotiable.... more