I got a used 8 port card from ebay for about $20 a few years ago. I had to flash different firmware to it using a DOS boot disk to disable raid though. It’s an obsolete card, but it’s been working fine in my NAS.
The temperature is set too low. The solder should melt almost instantly when applied to the tip if the iron is set to 350°C.
Those fine point conical tips that typically come with soldering irons are terrible. Get a 2-3mm chisel tip, that will be suitable for most soldering jobs.
I would just use a MOSFET and resistor to drive the LED if you are only using it in short bursts.
If you want to drive the MOSFET directly from the microcontroller pin, it will need a series resistor to limit the current since the gate has quite a bit of capacitance. If your microcontroller pins can handle 15-20mA, it will be able to switch an AO3400A fast enough without needing a gate driver.
Since the remote control signals are short and low duty cycle, you could use a capacitor to provide the peak current for the LED without going over the maximum current of the power supply.