Why are you considering an antibacterial laundry detergent? If you want to kill any bugs on your knicks, that is a waste of time because as soon as you put them back on, the normal bugs you carry on your skin will move back onto the knicks. If you are concerned about picking up bugs from other people using the facilities, then you need to make sure there is adequate contact time for the antibacterial to do its job and depending on what it is, this could be 10-15 minutes or more. I don't think that is going to go down well in a crowded camping site with tons of people queuing up to do their washing also.
I would suggest you take a collapsible bucket, that way you have complete control over the hygiene of your washing facilities and you can take your time and soak things well. Use the hottest water you can stand (if it is available) and a liquid laundry detergent as it dissolves more easily than powder, although it is a bit more of a nuisance to carry. Make sure you rinse your knicks well as detergent residue can irritate the skin.
I would suggest carrying a minimum of two pairs and wash each night, that way if one pair doesn't dry properly overnight, you have another pair to wear the next day.
If you really want something with antibacterial properties, consider taking a bar of antibacterial soap and use that instead of laundry detergent.