There are two times you may find yourself using the traffic pattern checklist and the caveats are different:
1. A Traffic Pattern.
We did these for a living in the Air Force, it seemed for every hour of operational we had an hour a training. As an instructor, you had to have it all together to get the syllabus requirements finished while keeping the airplane right-side-up. As a civilian Gulfstream 450 pilot, you spend very little time training and it is that much more important to have it all together before you blast off with a student in either seat.
Here is a video of a short pattern ride for an idea of how we do it at Incognito Air: Bedford Pattern. The ATC is clipped now and then to disguise our call sign and location on the field, we have to remain incognito, after all.
2. A Short Hop Between Airports.
If you are flying between two airports and will never go above 10,000 feet you may find the traffic pattern checklist to be advantageous. Simply use your normal checklists up to the Line Up, then the traffic pattern checklist through landing. After landing go back to your normal checklists.
What follows comes from the references shown below. Where I think it helpful, I've added my own comments in blue.
[G450 Airplane Flight Manual §2-07-10]
Prior to takeoff, as a minimum, you should have programmed the FMS to level off at the pattern altitude and briefed the pattern to be flown. If time permits on the ground, you should also program the arrival procedure and landing init. It would be helpful to have already briefed the approach and aircraft configuration.
For your initial takeoff, you should accomplish the normal before engine start, taxi, and line up checklists. You should commence the use of the traffic pattern checklist once airborne.
You should get the ground spoilers off first, or as soon as the gear is up.
I recommend setting 10° flaps after takeoff and selecting manual 180 knots. Once pattern altitude is reached and landing init is accomplished, auto speeds can be selected.
You need to ensure the student levels off, stabilizes the airplane at pattern altitude in proper trim. A common mistake is to level off without trim and enter a shallow descent.
If the student is behind or tends to give long, meandering briefs, you can do the briefing: "This will be a visual pattern, you will be keep the airplane at pattern altitude, extend 20° of flaps when abeam the touchdown point, extend the gear when ready to turn base, and enter a shallow descent. I will clear for traffic. We need to roll out about 2 miles from the runway about 600 feet above the runway. Extend 39° of flaps as needed for the descent, cross the threshold at VREF plus five, and land in the touchdown zone. Any questions?"
If you didn't get to this on the ground, as a minimum you need to enter the arrival identifier and do the landing init. Selecting the runway to the scratch pad, adding a "2" and entering that over the runway line will give you a two mile final.
Here is where more than a few Air Force instructor pilots made serious mistakes and lost perfectly good airplanes. You might try this technique to prevent forgetting something that can hurt you. Don't allow the student to depart pattern altitude or make the turn to base without the gear extended. Keep your hand on the gear handle until you have three green, no red, and have confirmed the weight on wheels system indicates "four in the air." Then arm the ground spoilers and put your hand on the flap handle. Don't take your hand off the flap handle until they are set for landing.
Check "four in the air" before . . .
If you are heading for another pattern, run this checklist. If you are done for the day, put this checklist away and return to the normal procedures checklist, beginning with the After Landing Checklist.
[G450 Airplane Flight Manual §3-18-50] Landing gear should remain extended until all brake temperatures are below 200° C. Verify Brake Overheat message is extinguished.
Even without hard braking, multiple patterns and taxi time will raise the temperature of your brakes. If you see more that 200° you need to note it and consider leaving the gear extended for the next pattern.
To save time consider: "This will be another visual pattern same as the previous, except..." and cover the exceptions.
It typically takes two minutes to fully program the FMS for a visual pattern, depending on your proficiency level. You will exhibit good flight discipline and will set a good example for your student if you ask ground control for two minutes at the end of the runway to set things up, rather than do all this heads down while the student navigates his way to the other end of the runway.
Gulfstream G450 Airplane Flight Manual, Revision 35, April 18, 2013