A battery is made up of individual cells, mostly 18650 cells. The 36v is made up (normally) of 10 cells in series. They are approx 3.6v each, so giving the 36v (though this will be 41/42v hot off the charger). Each cell has a rated current that it is capable of delivering without voltage sag. So, if the cells used have a current capacity of 5A, one string of 10 cells will deliver 5A. To get 30A you would need 6 strings in parallel. So this would be a 10s6p battery. Now, in practice, in order not to get voltage sag during prolonged high draw events (climbing hills) it's best to have a battery that is rated at twice your controller maximum current (or at least 1.5 times). So, in your case, this would be a battery capable of delivering 60A. With 5A cells this would be a 10s10 or 11p battery. That's a big battery.
The reason you don't want voltage sag can be seen in part of Ohms law:
P=IV That's Power = Current X Voltage. So if the voltage drops so does the power available. Really not useful when your legs a burning half way up a stiff climb. If the sag is really bad then the voltage will go below the low voltage cutoff (LVC) of the battery management system and the power will cut off entirely. When voltage recovers (once load is removed) then the system can be restarted. However this stresses the battery and if done repeatedly, will shorten battery life.
So what your installers are saying re voltage is true. But they are not asking or telling you that your batteries ability to deliver the current may not be sufficient.
It will work, but for how long and at what range?
As Neal said, your current battery will probably have been configured to supply the needs of your existing controller. It may well not be very satisfactory with the new system.
I have no experience with Cyclotricity kits but a 30A controller sounds very high to me.
Hope that helps.