There is not $869 of taxes. This is a poorly trained call centre staff fraudulently claiming the BA "carrier surcharge" is a tax. It is not, it's paid to BA. $194 sounds about right for the government taxes.
You could avoid the "fare increase" by shopping around for dates where the same fare class is available. Depending on the season you booked and whether you got a sale fare it would be possible that the same price isn't available -- however that seems unlikely. If it was a sale fare it usually wouldn't be changeable at all and November is hardly peak season. (And you're booking plenty in advance of your new dates).
You could also ask if you can cancel the reservation and pay the change fee only when you're sure what your new dates are and reuse the ticket. I'm not sure if BA allows that but other airlines do.
I would stick to your guns about the $275. It's their web site so it's not your problem if it's a glitch. You should say you relied on the information their web site provided and expect them to stand by their price quote. If they don't you might consider going ahead with the purchase and then filing for a refund with customer relations or even sending the screenshot to your credit card and contesting the charge.
Between all your eticket and reservation confirmations do you see the fare code anywhere? That'll be some gibberish like TKP5UK Perhaps someone here can look up the actual fare terms for your fare and you can find out what the correct change fee is. They'll need the origin and destination and perhaps also the dates.
You might also mention that this is all due to the visa delay. There *are* clauses in nearly all fares allowing flexibility in case a visa is denied. I think that's intended to apply to people who are refused a visa to their intended destination but there's a slim chance they might consider treating it under that clause.