I know they use big NiCads on airplanes just fine, jets, piston poppers, and turboprops. Some of the turboprops will completely drain the NiCad in about three start attempts, they get fully recharged in a minute or two once one engine is running, so if it is done correctly in a properly designed system NiCads can work. These batteries last many years. The trick will be getting the bike's system to act like it was designed correctly.
The little RC NiCads are 1.2 volts per cell, ten of them in series would make a 12 volt battery pack. Eleven of them would be 13.2 volts, closer to what the charging system runs at, would this work better than ten of them? Would some resistors or a rheostat in line with the batteries better control the voltage? Maybe an adjustable voltage divider to adjust the voltage at the sense wire of a Honda regulator? Would a separate additional regulator need to be designed to apply just the right voltage to the battery? Need a little help from our electrical engineer buddies. How would one determine what capacity each cell should be? Small ones would be easier to put on the bike, under the seat or in the tail or someplace.
The 550/560 hybrid clone cafe racer will be needing something like this.
Nicads will accept charge even more rapidly than previously mentioned, 3 or 4 C (15 -20 minutes from dead to full charge)(even considerably higher rates than that under some conditions), but I suspect they are not a good choice for your intended application (smoothing out the R/R output, but not starting the bike)
Although they will accept charge rapidly, once they are fully charged, putting any more charge into them damages them rapidly. The problem is that once charged, they "peak" and the voltage declines, which (with our style R/Rs) would tend to cause them to overcharge rapidly and be damaged (unless there were enough cells in series that they never became fully charged, which MIGHT be do-able ... I don't know enough to speculate)
If what you want is the "three start attempts" capablity described above, you could probably get that with NiCads, but you would need a special charger for them. (i.e. not our R/R system)
It may (or may not) be possible to get such a system to also do the "smoothing stator output" function as well.
But assuming all you want is the smoothing, I suspect that the way to get that would be either via capacitors or a
MUCH smaller lead acid battery.
Given that you don't intend to use the battery for starting, I suspect you could go much lower than 14 AH, possibly by a factor of 10 or more.
I have seen some very small sealed batteries used in alarm systems, which might work well for that. (battery volume and weight will scale with capacity, so less AH capacity will be smaller)
I don't know how well those will tolerate the output of the R/R system, but I suspect it will be better than NiCads.
The other alternative would be the use of capacitors.
This has definately been done, and apparently at least sometimes succesfully.

I don't know any details, though I remember occasionally reading in the forums here about people doing so.
On the minus side ...
I have read several stories on the forums here of people blowing the main fuse (which disconnects the battery) or intentionally running the bike without a battery (and no other changes). At least some of them ended up blowing out their igniters. (OTOH, others apparently managed to remove the battery and run the bike successfully without damaging anything as far as they could tell)
So there definately are risks.
I have no real answers for you, but some things to think about ...