What pace to run for LT 3 mile?

Your 5k PR would suggest you run at 5:38 pace for 20-30 minutes. However you said you're not in that kind of shape yet so I'd do 5:45-5:55 pace for a safe bet. Look at this way so you can understand why you are doing what you're doing:

93% (of your current 5k pace) for 20-30 minutes short tempo

90% for 40-60 minutes for a medium tempo

87% for 70-80 minutes for a long tempo (don't really recommend this as you sound like your in high school and this would not be very beneficial to you)

75%-86% is aerboic maintenance

and

anything less then 75% of your current 5k fitness is recovery

Hope that helps

One rule of thumb is to use your 10 mile race pace or half marathon race pace. Going off a 16:20 5k, McMillan says that would be 5:41-5:46.

Since you question your fitness, maybe do a step up - start at 6:00, then go to 5:50, then see if you are comfortable at 5:43

Update

I went out for the 3 miler yesterday and managed a 5:54 average pace. That's odd, given that I can run 6 miles pretty steady at 6:05 in training. I also noticed that my pace was all over the place. Ever since I got the Garmin, pace is constantly at a glance. I was fluctuating between 5:25, 5:40, 5:55... It was a windy day, but is this a cause for concern? Does this suggest I'm in about 17:30 5K shape? The effort was not all-out but it wasn't a walk in the park either. had I continued for another mile, I could have easily dropped to a 6:15.

No, not a cause for concern. My experience is that you have to do a tempo 3 or 4 times before you have a sense of where you should be.

The first time, you are worried about bonking, going too fast, how hard it will get in the last mile, etc. The second and third time you won't be so worried, you'll be in better shape, and will probably go faster.

on the loose wrote:

Update

I went out for the 3 miler yesterday and managed a 5:54 average pace. That's odd, given that I can run 6 miles pretty steady at 6:05 in training. I also noticed that my pace was all over the place. Ever since I got the Garmin, pace is constantly at a glance. I was fluctuating between 5:25, 5:40, 5:55... It was a windy day, but is this a cause for concern? Does this suggest I'm in about 17:30 5K shape? The effort was not all-out but it wasn't a walk in the park either. had I continued for another mile, I could have easily dropped to a 6:15.

Garmin units are not very accurate for instantaneous pace readings, but the average of a lap will be spot on. It's likely you were not fluctuating that much.