It might fit under the seat if you are careful. I only carry my big kit on the bike when I am going out of town and have bags. I keep the small "rope type" from the auto parts store in my tool kit for emergencies. If I am in town, I will arrainge to get my plug gun and do it right or use a rope and re-plug it when I get home.
On out of town trips, I usually coordinate with others on the ride if they are going. I have the plug kit. Someone else has the cooler. Another will have some snacks... Guys on goldwings have lawn chairs. Beemers carry duplicates of all of our stuff plus radar detectors, MP3s, GPS, compressor, spare gas, laptop, extra maps (including Topo maps), a novel, and emergency rations.