So far this seems to be the simplest solution: "Use a MOSFET to switch it but drive the gate high/low with a second transistor".
Will this work?
Yes, but it's not as easy as it sounds. Switching 3.5A at 38kHz needs careful selection of parts and getting the math right. The MOSFET will overheat if you don't.
Any more suggestions on how to make a simple prototype?
Maybe a TIP150 is better suited for high frequencies. It eliminates all the MOSFET switching speed problems and you end up with a simple heatsinking problem - much easier to solve if you're not an expert. If it gets too hot you can just run two of them (or more) in parallel.
You still need to find a way to regulate the current, too. You can use a resistor if you're not worried about power efficiency but when you do the math you end up with numbers like "1 Ohm, 50W". Such resistors do
exist (eg. http://www.ebay.com/itm/400310239042
) but when you see devices like that you should wonder if your cunning plan needs a revisit. There's usually a better way.
What exactly is it you're trying to do? A 38kHz IR signal is obviously for a remote control, but 100W? Are you trying to switch a TV on from half a mile away? A small radio transceiver would do it better. Are you trying to flood a football stadium? The light from a few dozen smaller LEDs would probably disperse better (LEDs are quite directional) and cause less lawsuits for eye damage.