The stock 5 ohm coils, take about 2.4 amps. The alternator draws about 2 amps. Different coils may draw more current.
The above is constant draw whether the engine is spinning or not, about 55-60 Watts, which is right about what you expect from the alternator at idle.
Electric start takes about 120 amps. If you still wish to electric start, I suggest you abandon your small battery desires.
That speaks toward capability. The next concern is duration or capacity. As batteries deplete, their voltage falls, as does voltage under heavy loading. If your bike starts quickly, 5-10 seconds is all you need. Batteries are rated in amp hours (AH), or how many amps it can provide in one hour and still have enough voltage to supply spark voltage. You will likely be best served with a battery rated 5 AH or better. And, if you aren't ever far from a shop battery charger, a 2.5 AH battery would run the two essentials for 20 minutes and still have enough to pop the spark plug... when the battery is new. However, such high loading and deep discharge cycles will shorten battery life, and require frequent replacement. But, for racing applications, that probably won't be much concern.
Cheers,