My employer has a similar policy, which basically involves our TA booking flights on Expedia, trains on the The Trainline, hotels on Booking.com and charging us a handsome markup for the privilege.
I've always told my employer that I am not happy to pass this extra cost on to our clients as we should be billing them as little as possible, and so will be booking it myself, I am happy to manage any issues that arise with that. So far, my employer hasn't pushed back on this, even though it is strictly against policy, so maybe worth a try.