Optimising a single pulse precision peak amplitude detector
Do you find any obvious mistakes that I am making here with my
component selection (or their values) or the circuit itself?

Looking at the 1N4148 data sheet, the reverse leakage current is this: -

I know R3 is there to bootstrap the junction of the diodes to reduce leakage current but given that the inherent diode leakage is about 1000 times more than the input bias/offset currents for the op-amp, I would want to make sure that my model and simulation do take this properly into account.

Is there a specific type of capacitor, I should go for?

Ceramics are normally the best choice so I'd go for NP0/C0G characteristics and read the data sheet carefully about what leakages may present themselves.

Is there a ready-made peak detector IC available already that would
perform better than this circuit?

It's an off-topic question but I don't know of any.
The 1N4148 is really good on reverse recovery time (4 ns) but make sure you test the diode model with it set to 10 ns just to be sure. There may be some subtle effect here that drains a fraction of a percent of the output voltage on low level signals.