Can't you get a fixed IP?
It's a vital service, that any ISP should be able to provide and should allow you to do what you please. If your ISP can't offer that, then I'd switch ISP's.
I hate using BT for home internet, but I cannot fault their business service, in fact it's been about 2 YEARS! since we had any downtime on our connection. It's expensive, but for good reason.
I would ask your ISP if they can give a fixed IP address to your router, tell them it's for accessing your home network from work, make out that your an IT engineer or something in the email, then they tend to mess you around less, you might even get straight onto the real techies who actually get stuff done.