Locum sounds dangerously out of date - been used since the 90s! Pioneered at St Marys.
Yes, continuing to take low dose (75mg or 150mg) aspirin is safe if advised by a Dr. Read Lesley Regan's book about it, it's the most common solution for repeat loss!
However regarding your previous losses if you have had a bfp and then a bleed, clinically that is a miscarriage NOT a chemical pregnancy and you should be referred to a Recurrent Miscarriage clinic not a CD21 test on its own. GP ignorance about this delays success for ages in some cases and makes me really 
. Essentially, a positive urine test happens 48hr after implantation: yes, emotionally it may be like a late period but crucially it shows the sperm/egg met and implanted, so the issue is with something at the next stage ... but the good news is often they suggest aspirin anyway for this.
What you can do is ask your GP (ask specifically for the proper GP or take the book and show the locum) if you can be seen by a consultant at your ante-natal (or ask your MW to ask her supervisor). They may wish to do a blood clotting test and adjust your dose.
I have had several very early losses and some later (not late, but later in 1st trimester) ones. With one of those, the gynie at the hospital gave me the little "if there's a bfp it counts!" speech and referred me to St Marys... my GP had been clueless. I am now at 18 weeks on aspirin and clexane, as my 'sticky blood' (mine is likely stickier than yours, don't worry! Most are OK on aspirin) had caused all but 1 loss (a partial molar) - especially, St Marys said, the ones that hardly got anywhere.
Long post but... (1) Yes, carry on taking as your GP advised, tell your midwife you've had several very early miscarriages and s/he advised based on this and (2) That locum is very out of date. Not your job, but I'd be tempted to contact PALS as aspirin in pregnancy is so common it really is general knowledge now (for GPs, I've found).