As always, I speak from personal experience rather than any in-depth expertise or inside knowledge; as always, I try to be pragmatic: as always, pragmatism isn't always the popular alternative.
With the greatest respect to Keith Beddoe, I humbly suggest that what you're seeking to achieve could, in theory, be done. I say 'in theory'; in the same sense that 'in theory', a simple, straightforward number port can be done. However, second only to matters pertaining to scam calls, since problems with number porting are one of the most common sources of discontent on this forum, it's glaringly apparent that the theory and the reality are often not the same thing.
A call-sign number is just a number, like any other. As a flow-chart, the process is simple: BT assign a number, any number, to a given line. People dial that number; it rings the phone on that line. The customer requests Call-Sign; BT assign a second number to the same line, but with a different 'ring' on the receiving end. People dial the number; it rings the phone. There's nothing magical about Call-Sign numbers; once a Call-Sign number is no longer being used, it can be assigned to any other line, either as a normal working phone line or as someone else's call-sign number ( I say this based on the fact that a number I used to have as a Call-Sign number is, I know, now someone else's main working landline number ). However, this is where theory and reality part company.
Assuming you're planning to stay with BT, and assuming you're moving to somewhere in the same area, you could ask them to port both your original landline number and the Call-Sign number to your new address. Alternatively, I'm not entirely sure what you mean by having a second landline installed 'here'; do you mean that your son is remaining at your current address after you move? If that's the case, for simplicity's sake, I don't know why you couldn't just sustain the status quo - i.e. leave the existing arrangements exactly as they are, the only difference being that for all practical intents and purposes, the Call-Sign number would remain your son's business number and would work no differently than it ever has ( although, as stated, that number will never display on outgoing calls from that line unless you actually go to the trouble of 'spoofing' it - which seems like taking a sledgehammer to crack a walnut ).
The upshot of all of this? My advice, for what it's worth - again, as someone who doesn't work for BT but has a great deal of experience with telephones - would be to simply bite the bullet. I know - believe me, I do - how galling it is when you realise some way down the line that you've done something the 'wrong' way; but the worst thing you can do is compound the error by going to extreme lengths to sustain a flawed situation which at some point in the future is almost certainly going to have to be put right in any case. If your son's business continues to expand and flourish as you and he presumably hope it will, the longer you leave it, the harder it will be to fix this. As big a pain in the **** as it is, it will ultimately be less painful and probably less expensive to get the business cards re-printed and the sign on the van re-done, than to try to sustain an ultimately unsustainable situation by one of the means you've suggested, even if that proved possible in the short- to medium-term. Notifying 300+ customers of a number change might seem like a huge job, and I'm not suggesting it isn't ( maybe a blanket e-mail ...? ); but come back to me in six months' time when you've tried going one of the other routes, spent countless hours arguing with BT or any other provider(s) about what they said they could do but haven't, and then let me know whether biting the bullet now would have been the better choice.
At least if you ( or your son ) deal with this now, and he gets himself a proper business landline, this won't be hanging over him / you with the worry that he's still going to have to address the problem at some ( almost certainly less convenient ) time in the future.
Sorry the advice isn't more cheering; like I say, pragmatism doesn't always tend to be popular.
