people not paying their licence fee puts up the cost for honest people who do pay it
I'd like to see your working for this incredibly dubious claim.
The BBC is notionally a non-profit organisation. It costs the same for them to produce and air a programme irrespective of how many people ultimately watch it, and since their UK viewing figure has no bearing on the return, it's entirely irrelevant how many UK people actually watch the programme. You are essentially using the exact same "lost sale" fallacy that software companies have always applied to acts of software piracy.
If people viewing the BBC without a licence pushes the BBC's costs up, then the same must also be true of people who do not watch the BBC and therefore do not require a licence, because the BBC still produces and airs programming regardless, after all, those are also "lost sales".
Except the "lost sale" thing is complete nonsense, because the bulk of those who view without a licence would simply stop viewing before they ever paid, the exact same way that people who pirate software would just go without if piracy was not an option. You can see this with software which can not be pirated. What happens, is contrary to this "piracy increases prices" lie, when companies have a monopoly due to the inability to pirate their product, prices rocket through the roof. See Microsoft and their OS prices.
"Stealing" from the BBC is in no way similar to physically shoplifting, and that is a completely asinine comparison. Goods in a shop have to be produced and purchased, the cost of which is recouped via sale. If the goods are stolen, the cost was still paid up front by the vendor, therefore they have to recoup cost some other way if they are unwilling just write the loss off. Yes, the BBC obviously incurs a cost when it produces content, however, this is produced entirely in the knowledge that "sales" are entirely speculative and impossible to guarantee. It's not like producing 100 bricks in the knowledge that you can sell 100 bricks at a set price, it's entirely in the lap of the gods how many people choose to watch TV output, and if that total is none whatsoever, you've still spent the exact same amount producing that content in the first place.
If your claim was in any way cogent, we could very well end up one day with Mrs Smith, 10 Smith Street, funding the entire BBC on her own because nobody else watches the BBC but she doesn't want to miss Eastenders, so the BBC has ramped her licence fee up to the total of the BBC's entire annual budget.