If you can't fit the whale with wings, turn it into a big balloon, reduce gravity or make the atmosphere really dense, then your options are limited.
The Antonov An-225 Mriya would be able to lift that much weight (maximum total payload 253,820 kg versus whale-weight of 173,000 kg so no problem there) - the question is, would the whale fit in the payload bay without needing to be chopped into pieces?
The aircraft has the spacious cargo compartment with length of 43,32 m, width of 6.4 m and height of 4.4 m
Antonov: AN-225 Mriya / Super Heavy Transport
So we should be OK on length:
up to 30 metres (98 ft) in length
It's hard to find data on the width of a Blue Whale's tummy, but if we assume it's no wider than the tail, then we should be ok:
The width across its tail flukes (flippers) is about 6m (20 feet)
and on height we do seem to be roughly in the ballpark.
So in summary, this method meets the criteria in the question and should work OK, although the whale might have trouble squeezing its waistline into the plane - it might need to go on a diet first. If you could choose a slightly smaller whale, that would be helpful (173,000kg is particularly large, even for a Blue Whale).
You would need some way of helping it support its own weight out of the water, of course... (perhaps you could use the additional available payload weight to surround it with some kind of gel?) and some kind of a brain-computer interface, assuming you want the whale to pilot the aircraft itself.
Alternative answer: [drop it from a really high place][3].