There is little/no chance that BA will through check your luggage. Of course, you can ask but likely 99% chance the answer is no.
By October things should be fine. Flying out of Amsterdam in J is not an issue. I’ve flown out many times with BA during the mayhem of the last few months. I’ve never had to wait more than 20mins for priority security. Especially since you’re flying in the evening (if I understand you have a 10 hour connection?) I would check the bags in after picking them up and go enjoy Amsterdam for a day. The lounge (in the Schengen area as you’re going to HEL) is basic/mediocre at best.