I finished modifying the remote using the original IR LED, about which I know nothing, the mosfet with gate diven through a 470 ohm resistor, and current limiting resistors yielding 5 ohms. And I changed the big capacitor to 1000uF, 6.3V. The current is just under 500mA, and it works fine from my office, which is what I needed. The only thing I did that's a bit sketchy is to use two 1/4 watt 10 ohm resistors in parallel to get the 5 ohms. The voltage across them is about 2.3V, so they dissipate about a watt. But it's 38K with a 30% duty cycle, and it only sends one command at a time - the Mute button code - so I think the resistors will work ok.
I programmed the Pro Mini so if I hit the "*" button the remote will simply transmit the Mute code and go back to sleep. If I hit the "#" button, it will toggle the On/Off code, and go back to sleep. But if I hit a number key, or multiple number keys, constituting a numerical value, it will send the Mute code, wait for a number of minutes equal to the entered numerical value, then send the Mute code again, then go back to sleep. In all cases there is a three-second delay before sending the code so I have time to return the remote to its cradle, which has it pointing directly to the TV.
Since the voltage across the resistors is 2.3V, there's room to add a second IR LED in series, and double the transmit power, if I ever need that - well, as long as the battery is pretty fully charged at least. But it works well enough for now.