Can I draw brief large bursts of current from a shift register, above output pin current rating (i.e. cap or IR sensor on the register output pin)?
What are absolute maximum ratings there for? No, what are they really there for?

They are to protect the manufacturer's reputation. Engineers talk to each other. I used this ACME part and it failed. I used this NXP/TI/Analog/Maxim part, and it kept working! The reason it kept working is because the data sheet accurately reflected what the part would do, and warned me against trying to make it do things that, in the long term, would cause sufficient numbers of devices to fail, that the manufacturer would get a bad name for selling things that didn't work.

So, can you draw 50mA from a pin that's rated 20mA absolute max? Certainly. For how long? Maybe a few uS, maybe a month, you do the tests. The manufacturer has already done the tests, thousands of hours of them, and decided that something about the part, the bond wire fusing, or electromigration in the diffusions perhaps, means that they expect that running at that current will change something vital about the device, like how well it works, or whether it works.

As you are anticipating pulsing the diode, putting a capacitor across the diode is the worst thing you could do, as it increases the inrush current when you pulse the diode on.

Use the limited current from your S/R pin to switch a driver that is specified to handle the IR diode current.