The instructions on the Brightwell product say "ADD one Hardy Fish" (not Dr. TIms Ammonia) and then test for ammonia like you would for a fish in cycle. Dr Tims will provide ammonia much faster than a hardy fish will, so I suspect it may work, but it doesn't match what the instructions say, ("Hardy Fish">).
Some people thing fishless cycles (DR Tims Ammonia) are kinder and gentler, but I think the bacteria might beg to differ? I would use the Dr Tims and then measure the cycle per the bottle instructions, which is ~ "wait until you can't detect ammonia and nitrate is very high," ie a normal cycle process for making a tank safe for fishes.
I think it should work fine.