I don't fly myself but I've done quite a lot of reading on TCAS theory (formulas, thresholds. operations and stuff) recently, so I'll try to answer that one based on pure theory.
For a TCAS RA to occur, both TCAS (slant) range test and TCAS altitude test must both be fullfilled.
D_mod for the range test at such low an altitude is about 0.35 NM which is equal to 2000' (slant range might also be pure vertical). The altitude test should be triggered when your aerobatics mate climbs through (roughly) 4300', that's 20 seconds away from climbing through 6000' (asuming still 5000 fpm at 4300'!). With such high a climb rate, TCAS II is unable to get a sufficiently good estimate for the vertical seperation and assumes 0' of vertical seperation, to be on the safe, conservative side.
I'd say this would quite likely trigger an RA in your Boeing, as both Range and Altitude Test are triggered. However, this is a pretty unfortunate conflict situtation where TCAS - to my best knowledge - is known to perform less than optimum (but still safe) against aircraft at slow closure but high vertical rates with no Mode-S transponder.
TCAS formulas choosing RA strength and direction are a bit difficult, but I'd guess that TCAS II chooses the non-crossing RA, e.g. a 2000 fpm climb, which might achieve a 600' vertical seperation.
In short, TCAS might work not too well in this encounter and it might be indeed be suitable for your mate to switch of Mode-C as long as he is doing aerobatics below class B/C airspace, and ensures staying clear of this airspace. This might still trigger TAs in your Boeing, but not RAs.
However, other guys from the front office might want to comment whether this is an acceptable advice from the flight deck's point of view.
Hope this helps,
EoD