I used to use a Tuna Melt, but replaced it with the Cool Cat Tremolo (same basic opto circuit I believe, but true bypass, and much more positive footswitch operation). I have never noticed the problem you mention, but (1) I always use quite a fast rate, and (2) I rarely if ever use drive at the same time (but if I did, the drives are all before the trem).
If you are using a slower trem rate, then occasionally the attack of your note will coincide with the 'trough' of the trem, and you will get a slow-attack thing going on. I guess this will be more noticeable if you are expecting your sound to cut through because you have stomped on the drive pedal.
‘There is no excellent beauty that hath not some strangeness in the proportion.' - Francis Bacon