So Vic0239 is saying pretty much the same as me, i.e. Gigaset N300ip does what you want - probably....
I've just switched away from my N300ip as the BT landline finally went totally dead.
Not sure if I'm allowed to say this on here but it's packaged up with a Gigaset R630h Pro handset ready for sale on eBay when I can be bothered to get round to it. Both fully reset. Happy to consider a private sale at a mutually agreeable price.
Note that the N300ip will happily register and support most GAP-compatible DECT handsets, indeed I used mine with several of different brands initially, but you really need Gigaset handsets for full functionality, especially if you want to use multiple numbers at different phone services. Though the web control interface to the N300 does everything you need to set it up from a PC.
I've only used AAISP for VOIP and BT retail was my POTS landline supplier. I have two AAISP numbers and did have one BT one until recently.