The "Host +5V" output is specified for 100mA load maximum. I do not know how or why the current is limited but I do not think that it is a good idea to operate the NI card out of spec.
If you need a voltage level of more than 2.5V to trigger your device, and if you are already using a 1kOhm resistor from the trigger input to GND, connecting the "Host +5V" output to the trigger input via a 470 ohm resistor will show a voltage level of approx. 3.0V on the trigger input. If you close the switch, the maximum load on the "Host +5V" output will be a few mA, well below the maximum specification. So replacing the 1 ohm resistor (this value barely makes any sense) by a 470 ohm resistor probably will solve your problem. To avoid any problems due to faulty circuits or components, I would recommend using a 100mA fast blow fuse as close as possible to the "Host +5V" output.
Anyhow, there might be some glitches/spikes on the "Host +5V" output during powering on and off your host PC, so maybe the voltage level on the trigger input will be reached for very short periods before the "Host +5V" output is stable. If this might cause any problems (on our systems it does!), you would be better off using a "true" reset controller which has some hysteresis for voltage level detection.