It will not be lower than 35kA, and may be higher.  Other than that there is not enough information available.

If you have a transfer switch with a "kAIC" rating, it must be one of the types that utilize circuit breakers as the switching devices. You should be able to read the circuit breaker part numbers somewhere and look it up in the mfr's data sheets.

If the transfer switch mechanism does not use circuit breakers, it does not have a "kAIC" rating on it. The "IC" means "Interrupting Capacity" and if the devices are not fault interrupting devices, such as circuit breakers, then they cannot have a kAIC rating. If someone is telling you otherwise, they are mistaken.

If this is a MANUAL transfer switch, i.e. a double throw safety switch or a set of interlocked rotary switches, and there is a set of fuses ahead of each side of the switches, then the fuses will have kAIC ratings at the various voltages. But technically, the switch itself does not have the rating, the fuses do and the switch has a "withstand" rating, now also known as an SCCR (Short Circuit Current Rating).  

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Here is the link of transfer switch:
http://ascouniversity.com/Products/ATS/series300se/300seratings.html

This is a service entrance type transfer switch, so it has a circuit breaker as well as a switch. The non-service-entrance type transfer switches don't have a circuit breaker (they only have the switch).

I think if I need to put a non-service-entrance type transfer switch; then I will have to put a separate upstream circuit breaker (in between utility and transfer switch).

Is it correct?

All the information you seek is in the brochure on that link. You might want to read it.

It appears that they use Sq. D. circuit breakers, so you can look up the AIC ratings of those at 240V, but I think I saw that specific info in the brochure anyway.

If you use your own circuit breaker, you will need to know  the withstand (or SCCR) rating of the switch when used with that SPECIFIC breaker. If you use something that their switch has not been tested with, then you will need to limit the fault current to whatever level they have WITHOUT a breaker, which appears to be very low.

I would use the version with the breaker and listing already done if I were you, rolling your own has become a major headache now with inspectors.

 

"Dear future generations: Please accept our apologies. We were rolling drunk on petroleum."
— Kilgore Trout (via Kurt Vonnegut)  
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