Q: What is the reason Lithium Polymer batteries are limited to 1C charging?
A: They're not, at least not fundamentally. You can get 5C and 10C charging batteries from e.g. A123 systems. The reason batteries are limited on paper to relatively slow charging rates is to avoid internal hot spots, caused by a chemical-physical phenomenon that causes lithium chemistry batteries to discharge relatively evenly over their electrodes, but actually concentrates charging current to the areas that have the highest ion mobility. This is chemistry dependent; there are cathode materials that avoid this problem mostly and thus can charge faster. It's fairly specialty stuff though, and it is generally not a good idea to use chemical energy storage for rapid-cycle applications. I will come back to this later
Q: If I'm charging a Li+ battery for 90s at 1C, can I safely charge at more than 1C?
A: Most Li+ chemistries can safely charge up to 2C with a voltage cap, but with reduced lifetime. Not deeply discharging the batteries will alleviate this though, and then some. The cycle life of a battery increases at least tenfold if you discharge it at most 10% instead of 50%. Check with the manufacturer or datasheet to make sure though. If there is no datasheet, you shouldn't use the battery in a production environment anyway.
It's not a good idea to use a battery as a quick-cycle energy storage. Even with shallow discharging, especially high current batteries will only last you about 10 000 cycles. This is what (ultra)capacitors are for, especially if you are oversizing the battery anyway. The only downside to a capacitor is the ~10x decrease in volumetric efficiency compared to a battery. Without that problem, capacitors are much more suitable for quick charging and discharging. You can charge Maxwell's BCAP series within 15s, for instance. That's 1000 and 3000F capacitors, hundreds of amps. Besides, with large charging and discharging currents, your supporting circuitry and cabling will probably be larger than the power source anyway.
Also, it's generally a much better idea to store and use lithium chemistry batteries at the top of their capacity, near 4.25V. You get more power out of them for less current and you charge them faster with lower C-rating. Also their apparent internal impedance is much lower the 'fuller' a Li+ battery is.