You have posed a complex situation where multiple things are happening, I will try to break into pieces.
Say your substation has a low voltage condition, for whatever reason, then you are correct, in that the low voltage on the primary will mean a low corresponding voltage on the secondary. The voltage will be further decreased during this large current draw, due to additional current in the line; due to voltage drop being dependent on current. At some point we have to assume that the system stabilizes under this condition and the substation can provide an acceptable level of voltage support.
But, the load (secondary of the transformer) will draw more current, and the transformer is still going to deliver the higher current to the load. Transformers are a strange beast because they can be overloaded for quite some time, before heating takes a toll on the longevity. (some sacrifice in voltage regulation does occur). Under steady state conditions (i.e. loads already running) as the voltage goes down, the current will go up, because the load is based on KVA. Remember the rating of the load cannot change, it is fixed, therefore the current and voltage are inversely proportional. What may occur at some point, operating the load at the higher current for a long time may cause the longtime protection to trip. For motors, that would be the overload device(s).
Now, that was steady state operation, prior to the fault. At the time of the fault, the lower pre-fault voltage on the primary line, will cause less fault current to be able to flow to the primary side of the transformer. (less pressure to force it down the pipe) This is analogous to motor starting, when starting at a lower voltage means lower current draw (i.e. soft starter) because during starting the motor impedance is predominantly reactance. However, because of the inductance in the transformer, the transformer is already limiting the fault current to the secondary side. The amount that the transformer is limiting the secondary fault to, is in most cases, less then what would be possible even if the primary line was at a normal voltage level.
So, in general, knowing the service transformer size and impedance should suffice to get started. In all cases, you should ask for the utility information, if you can get it that would be a plus. Certainly, if anyone suspects unusual circumstances, then the more information you have the better. But utilities usually will not perform a system study for each service installed (costs to much). What you may be able to get is the system planning information, but that probably won't do you any good, because it is based on a lot of system growth over 5 or 10 years, and I think you would find that the infinite bus would still be were you end up.
Obviously, if your building a substation, distribution system, or putting in a very large service, they are going to provide more information, but in those instances that would be the engineer's job to determine for you.
Keep in mind that the cable has not been mentioned in what we have talked about. Cable has impedance, so just like a transformer, it will choke the fault current down. Therefore, it is important to include it in your calc. Cable has more of an impact on decreasing fault current then people realize.
Sorry for being so "windy" but I wanted to try and explain it right.