Whatever you do, don't get the MOT garage to do the work, as they will charge you an absolute fortune. I made this mistake with my Fiesta when it failed MOT (back when I was ignorant about the prices of car repairs and how easy they are to do yourself), which needed (in retrospect) about £200 worth of work doing to it, at the most. However, I made the mistake of getting the garage to do the work - and was landed with a bill for £900. The car was worth £400 at the most. So I told them they could keep the car, as it was worthless and the work was just too costly. That same garage has now closed down, incidentally.
And £150 for a backbox? Sounds a bit steep, a stock unit should set you back no more than £100. And fitting new handbrake shoes should solve that problem yes. However, sounds like the garage is talking ######## about the brake pipes - it can easily be done yourself, or with a mate if you don't feel comfortable doing it yourself. And as for a garage having to "make" the brake pipes? No. eBay, or Euro Car Parts will sell them for pennies. Sounds like the garage is just trying to take your money fwn