Is it not the case that if you OLCI but not print your BP then once you get to the airport you can then still change your seat for free using the SSCI machines (when they are available of course)? 8 months or so ago I did this for my daughter (she has no status) after BA had allocated her a random middle seat almost at the back and I changed it on the SSCI machine to an aisle in about row 12.

Plus my OH (no status) always gets allocated a random middle at the back but when she arrives at the airport (it has no SSCI machines) she goes to the BA check-in desk and asks for an aisle more towards the front. They always change it for her....at no charge ^

So you may have some options to change it for free....