54 pounds, plus R134, plus evaporator, plus hoses, plus mounting brackets, plus whatever the core weighs. You are looking at 100 pounds of stuff. I agree, it's not a huge deal for a non-race car. But it is the weight difference of an automatic versus manual transmission.
The car shuts off the stock compressor anyway. The added compressor will put a strain on the alternator, so that's something to consider. If you increase electrical load, you increase the power consumption of the alternator.
The intake flow rate is pretty high, more than a third of a kilo per second at WOT in high power conditions. You need a high efficiency, high output system to be able to sufficiently cool flow like that. You can suffer a low efficiency system, when time is a luxury. In this case, it's not.
When you are attempting to cool rapidly moving air, you need to have a much colder exchanger. An accumulator, like Sumfoo1 mentioned, is your best bet for "blast chilling" the intake air.
You are talking about a system that will "super cool" your intake. Since you have transient lag between changes in temperature with any HVAC system, doubly impacted by a variable cooling load requirement, the bulb-feedback system is going to be inadequate for sufficiently controlling temperatures with any degree of accuracy. Your house doesn't go from 70 degrees to 100 degrees in tenths of a second.
I haven't, and I probably wont, do any calculations to progress this idea. SumFoo1 would be a better source for that, since HVAC is what he does.
As an engineer, I would invest my time in developing a properly designed AWIC. From that, you could have a small compressor running cooling pipes through the storage tank, to regulate the water mass temperature. If you want to design a "race" mode, I'd run a 20lb CO2 or Nitrous bottle, with charge pipes through the storage tank. The vented gas could be used to cool the exchanger up front (or maybe strap a kazoo to the end, so people will know you are serious).