I'm new to BKPs myself and wouldn't dare dream of advising you on a specifical model to get, but going by your description of what you want - and knowing the H-1001 well, as I own three of them - I have to say that honestly, you can't get much "tighter" than an EMG 81 already is, especially in the 1001s which have a more clinical tone than the ESP Standards anyway.
Problems with mixing passive and active pickups aside, precision is what actives were made for and the 81 is the king of them. Of course lots of people don't like that dead, flat response from them, but that's the price you pay. They're as clinical as can be, it costs you tone but if tightness is what you want while chugging away on a down-tuned metal riff, that's what the EMG 81 specialises in.
Again, I wouldn't want to give the wrong impression about BK or anything, but it is pretty much a fact: passive pickups have more tone and sound lively, active pickups (especially the 81) have more precision and sound dead.
One thing I've found with the H-1001 is that, I think due to the shallow break angle over both the nut and bridge as well as the obscenely thick finish on the neck, excessively low action results in a very flubby, damp sound, regardless of the pickups (I've noticed this with the EMG 81, 85, 60A, SD Custom, DiMarzio Evo 2, PAF Pro and Swineshead Warthog, so I believe it has more to do with the guitar than the pickup used). Raise the action just a fraction and suddenly it becomes much clearer. Don't underestimate what the right set up can do, especially with guitars like LTDs which frankly never arrive in proper playing condition. String gauge is also very important with active pickups and seems to make more of a difference to your tone than it does with passives.
For what it's worth, the action on my H-1001 sits at 2.2mm at the 12th fret (with .11-.56 strings tuned to Drop C), still low enough to play quickly on and sounds very crisp - currently it's got an EMG 85/60A combo in it, with an 81/85 combo it would sound even more sterile (perhaps even too much). When I did have the EMG 81 in it, I found using an EQ pedal set to a 'W' pattern (boost the extreme lowest, extreme highest and dead-centre frequencies) helped bring the pickup to life a lot more. If you do use active pickups then you need a good EQ pedal. It won't give you the depth of tone that passive pickups will, but it does go a long way in making actives sound a little nicer to the ears whilst keeping that signature active precision.
To add something that perhaps won't get me quite so brutally murdered by the BK illuminati, I know that Whitechapel use EMG 81-7s and 707s (essentually the same as the H-1001's stock 81/85 set), although a Google search tells me that Periphery use BK Painkillers and Warpigs in the bridge positions of their guitars, plus one of them uses a Cold Sweat in the neck. I myself am intending to switch the Evo 2/PAF Pro combo in one of my H-1001s to a BK Aftermath/Cold Sweat combination (based on what everyone on here says about them), hoping for a slightly more open and dynamic tone.