Hi Hevreh,
I presume that your weakness is one-sided, present on the AN side. I am writing this answer assuming that this is the case.
Your neuro-oncologists are absolutely wrong: Pt or at-home exercise will help. Maybe, they are just ignorant of the benefits of exercise during the recovery process like a lot of doctors, or they just don't want to write a prescription for PT thus admitting that their treatment led to your problems. I don't know their motivation but I vehemently disagree with their position on exercise.
Let's take your problems one by one:
balance - when neuro-muscular weakness is present, your balance problems have two sources. One is the balance nerve itself which a problem all ANers have, that is one side is missing or disordered balance input which all other people normally have. The way to solve this is to take vestibular retraining, physical therapy specifically aimed at at restoring your sense of balance. I recommend that you take at least a few sessions, and if this proves to be too expensive , continue at home. Usually, when balance nerve is not working or missing, the body compensates by reading about its position in space from proprioceptive receptors in arms and legs. However, if you have weakness or numbness there, this job becomes more difficult because the overall level of activity is reduced, thus creating a vicious cycle - improve you neuromuscular weakness to improve your balance but you need your balance to improve your weakness. Plus, anybody with one-sided weakness will be thrown off balance even if their balance nerve is perfect. I am just saying that a combination of AN and one-sided weakness spells a very balance for a while!
weakness - the way to improve it is with weights. Anything will do - equipment at the health club or dumbbells and ankle weights at home. The most important thing is to do it regularly 3 times a weak and stick to it. It absolutely works, but you have to do it for months and maybe years. I know I have to do it forever. The moment I stop the exercise, my weakness returns.
When you are working on the machines at the health club or doing floor exercises at home, you don't tax your balance nearly as much as when you are trying to walk or garden, let's say. So, by doing some sort of weight training, you are actually helping your balance in addition to weakness.
Additional things that improve your balance are Pilates, yoga, walking, gardening and , believe it or not, housework, like bending and picking up things or mopping floors.
Seriously! The last one is cheap and effective and cannot be called a hi-tec solution! I also found that hemiparesis (weakness) creates bad criculation and foot massager is great. I am going to Paris this summer and despite the weight restrictions on the airplanes, I am packing my 5-lb ankle weights with me first. I plan to do a lot of walking and don't want my leg to give out and start hurting. I am probably the only person in the world who takes the ankle weights with her to Paris but I need them to enjoy my trip.
I know you are worried about what to tell your family. You can tell them to expect to see you tired, less active and needing frequent breaks during your recovery. I honestly believe that this is not forever. Most people recover, even if it takes a few years, and return to a decent functional level. Expect this to take a while and keep a consistent exercise routine along with good diet and good sleep. All the best to you and let us know how it is going.
Eve