My Trinket Pro waits about 10 seconds upon booting before starting the program that I have loaded on it.
Can I avoid this delay and the bootloader taking up space if I flash my program directly to the Arduino with ArduinoISP?
It looks like the Trinket Pro has an Atmega328P processor. You can change the fuses to not run the bootloader, in which case you don't need the space taken by it.
I have a hex uploader sketch which will upload a .hex file from an SD file directly to the chip, removing the need for a bootloader. Using ArduinoISP is another approach which will probably work. I'm not sure whether that fixes up the fuses for you.
If you just want to avoid the time (and not the space - the bootloader is probably only 512 bytes) you can use the sketch I linked to (without needing an SD card) to make fuse changes.
See Engbedded Atmel AVR® Fuse Calculator for fuse calculations. Basically you need to toggle the BOOTRST fuse (the low-order bit of the High fuse).