Here is the situation:
Vintage motorcycle, custom wiring harness (the harness is just on paper at this point).
Here is the problem:
12V, 7AH battery, ignition switch rated for 5AH (that needs to kill ALL power), horn that draws 5.5A at 12V.
Do I run a 5AH battery and underpower the horn or run a relay off the ignition switch (and how would I power the switch).
Unclviny
(Electrical moron of the highest Order)
You’ve got your units all screwed up, so no wonder it doesn’t make sense. AH & A are not synonymous.
A 7 amp-hour (AH) battery holds enough power to (theoretically) put out 7 amps for one hour or one amp for 7 hours or any other combo that multiplies to 7. Including 700 amps for about 1/100th of an hour. Although something will melt if you try to discharge it quite that fast.
An ignition switch cannot be rated in AH. It must be rated in Amps = A. That is the amount of instantaneous current flow it can pass without unduly heating. You can put a 2000 AH battery from a Peterbilt behind a 5A ignition switch, as long as you don’t try to pull more than 5 amps through it at any time.
Strictly speaking, putting a 5.5A horn downstream of a 5A ignition switch is a bad idea. As you’ve noticed.
Practically speaking, the horn is used in 1/2 second spikes once every few hours of riding. If the only load running through the ignition switch was the horn, you’d be fine. The overload would be 10% very briefly and very rarely. That would work for years with no danger or degradation.
But reality gets more complicated. When the ignition switch is on, all the electrical system is powered. The headlights. The tail lights. The turn signals and brake lights. The lights in the instruments. The ignition system itself; points, coil, etc., or a modern ignition module. Fuel injected bikes have engine control computers about like cars do. And all of these things consume power.
A vintage motorcycle probably won’t have all those things, but it’ll have some of them. And if you’re simply trying to wire all of those things directly downstream of the ignition switch with no relays, you’re probably gonna exceed the ignition switch’s 5A rating before you even connect the horn.
The typical thing to do is identify the biggest continuous draw item (e.g. headlight+tail light), and put it/them behind a relay. That ought to get you enough room to power that relay and everything else through the ignition switch. If not, find the next biggest load and put that behind a relay. Lather, rinse, repeat.
Some points:
The battery capacity doesn’t matter. You can put a 100Ah car battery on your motorcycle and it will run fine because the current draw from the battery (A) will be the same.
2) The killswitch rating should be in A not Ah. 5A is low. Assuming 300-350 watts @ 12V that means a max current of 25 to 30A. That should be the switch rating.
I searched eBay and there are several motorcycle killswitches for sale but they do not mention the ampere rating. Unless your bike has 100 lights and a 1kw stereo system you should be fine with one of those 
Not quite. Amp hours defined
It is the discharge rate over 20 hours not one hour.
A 7 Ah battery can supply continuously .35A for 20 hours.
Getting back to the OP, the switch is too small.
The solution being buy a higher rated switch or run the horn on a separate circuit powered through a relay turned on by the first switch.
Just checked on my bike (2008 Honda CB600 Hornet):
The killswitch doesn’t kill the lights, instrument panel or horn. Just the fuel pump and starter motor (and maybe ingition). And I doubt it cuts the actual currents for the starter motor, probably just the relays.
So I guess a 5A rated killswitch should be more than adequate for cutting off a couple of relays.
Plan B:
Vintage motorcycle (1971 Norton)
210 Watt, 3 phase alternator
Custom harness (- ground, all 16AWG)
5.5AH battery
10A keyed ignition switch
Horn that draws 5.5A at 12V
Dual output coil (it’s a 2 cylinder)
Electronic ignition
LED brake/tail/license plate lights, instrument lights and high beam indicator light (nothing else).
I’m “good to go” right?
Unclviny
Correction to your correction. That’s not how it’s defined, that’s the standard way of measuring and reporting it. The definition is still as the previous poster said, Amps times Hours, regardless of draw rate. The reason there is a standard, is because like any other electrical component, you will get a somewhat different result if you measure it a different way. Being less than a theoretically “perfect” component, 1 manufacturer could tweak it to give better performance with high current, another could tweak it for low current, they both would report it as the same AH rating, and neither would actually meet the claimed spec when using the standard test. The standard forces them to all stay on the same page. Then it’s up to the consumer to choose a high current or low current one for his particular application.
That’s not too far off from the starter current draw for a car with a large engine, though, since 1HP requires ~62 A from a 12 V battery. The ‘maximum’ rating for a battery is usually given in terms of ‘cranking amps’. Unlike Amp-hours, ‘cranking amps’ are defined with a specified measured value - the highest level of current that can be sustained for 30 seconds at a specified temperature (e.g. 0º F for ‘cold cranking amps’) while maintaining a voltage per cell of 1.2V (car batteries have 6 cells).
This is something you may want to check on for your battery, unclviny (assuming this has an electric starter). I imagine you’re probably within the limits, though if it gets extremely cold where you are you should know.
No Electric starter and I live in Houston, Texas so cold is not a big problem.
I could go with either a 5.5AH or a 6AH battery (my plan is to use the smallest battery I can but I’m a “horn-guy” and I don’t want to underpower it).
Unclviny
The 700 amps? My experience is that V-8’s typically have starter draws in the 100-150 amp range. I don’t recall ever seeing over 200 amps unless the starter was faulty or the engine seizing up.
I have had a couple of MC switches bite the dust, one ignition, one headlamp switch. The headlight switch was only available as part of a harness assy, so fixing it was painful and expensive. I would put relays on the horn, and one each for high and low headlight. I’d also run AWG12 rather than 16 on those circuits…the lines to the relay coils can be 16 no problem.
Beyond the bulb rating, there is the cold filament inrush current, which is ~10x the hot current for a few milliseconds. This is what eats the switch contacts.
The ignition switch died at high speed while passing on a pitch black night. Lost ignition and all lights. Very exciting.
I consider order of magnitude “not far off”. Your estimations may vary.