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Is this the shot you were mentioning? I sometimes use a thin felt strip on the bottom head of the cocktail drum and sometimes not. In general, any drum smaller than 16" will give you a terrible sound when using a felt strip as a muffle- 16" is big enough if the felt strip is very thin -but just. I seem to have slunk away from using felt strips on drums under 18" without even noticing I did it. So as I said before- the muffling is mostly a cleaners bag and the o-rings on top. If you tune the cocktail drum to the same low and muffled pitch you can get a nice double bass drum illusion going playing alternating notes with right hand and foot- fun stuff live... I have even thrown a small piece of light weight cloth, the size of a hand towel, over the top head and played it that way - good muffling on the bottom and a thump bass drum like effect on the top (hence the double bass drum feel).
14x14 cocktail bass drums are trickier to tune and sound good. In general- always mic them when playing live- like with a cajon, you can let the mic and PA effects give you a very big sound without have to bring the big drum. I suspect you've heard cajons mic'd that sound like thunder- it's the same idea.
You asked about positioning my drum - I sit quite high and like the snare almost waist level (I play trad grip)- so everything is up pretty high and positioned flat/level and I play down on them- nothing, no cymbal or drum is above my stomach level. You'll have a different feel for your set up I'm sure- good luck with your shows!