I don't know anything about a 5 degree bank angle limit for avoiding action, but on the 757/767 the TCAS assumes that the bank angle "will be less than 15 degrees for climb performance purposes".
Whilst we are talking about avoiding action (I admit that the following is mainly for the benefit of TMA controllers). When we are slowed to our minimum clean speed (close enough to holding speed as to be no different), a combination of a 30 deg banked turn and g loading and/or turbulence may result in a genuine stick shake warning. With this in mind, when you give an avoiding action turn, the actual turn itself will not be appreciably tighter than those you normally see.
The flap speed schedule only allows for "an inadvertent 15 degree overshoot beyond the normal 25 degree bank angle". We also get a "BANK ANGLE BANK ANGLE" warning when the bank angle exceeds 35 degrees. The up-shot of this is that to keep our heart rates to acceptable levels and to avoid bending the aeroplane we will not be ablt to turn much more quickly than normal in a TMA environment.
I actually had an RA not that long ago, and the concentration required to switch from calculating heights and ranges in order to follow a nice CDA to suddenly fly the aeroplane out of the (growing) red sector was quite stunning. All good fun, but I can categorically state that avoiding action in the horizontal plane is the only useful response from the ATCO and even then, with the flight director showing turn left, the controller saying turn right, the clearance requiring a descent and the TCAS requiring a level off - there was a lot going on!
G W-H