I thought it might be a good idea for those with the BBE to help those attempting it in a thread like this. Once enough information has been accumulated it could be cleaned up and put on the Wiki. I'll start with some basics.

• Skip all Cutscenes, they add to your total time.
• Tranquilizing a Frog while on a wall will cause her to fall and die, but will not count towards your kills.
• Ammo cannot be purchased for non-lethal weapons in The Boss Extreme, so stock up on Emotion and Anesthetic rounds in your previous run.
• The Solar Gun is invaluable during the Eastern Europe Chase sequence and your fight with Vamp.
• Raging Raven can be 1hit K.O'd in her Beast form with a fully charged Rail Gun hit to the head. A body shot will do 75% of her health.

• Auto-aim is useful for the bike chase, as are smoke and flash grenades for the sections where the bike stops.
• You can kill the bosses in their Beast form, as this won't count towards your total kills. When they are out of their armor, use the solar gun on them to down them quickly.
• Stock up on chaff grenades before the run. Having about 5 should be fine.
• Try and find as many batteries as you can also, as this is very useful for using the Solid Eye for a long time. They can be found in the briefing section of each act if you explore with the MK.II.

Good tips. Also Stun Grenades very quickly kill the Beauties, and they can't dodge it. There's nowhere else in the game you have to use them, so it's a good option if you can't recharge your Solar Gun.

Doesn't using the Rail Gun on the  bosses cause a kill count?

Not if you "kill" the bosses in their Beauty form by non-lethal means.

oh damn, that's good to know... I've always thought you had to kill both beauty forms and beast forms non-lethally

Yeah, they can't technically 'die' in their Beast form.