Bagel is annoying as shit as melee. With Insect Glaive it's not AS bad because I can do some range pokes and stuff, but on the flipside I can't fucking figure out how to get orange buff reliably, so no super armor/earplugs/5% defense buff which sucks.
For the double tempered Bagel what I did is mostly sit back. It took a while, I just literally sat there and watched them kill each other, with the cat throwing molotov at them every now and then and using my kinsect on their tails to cut it.
As melee, your first priority should always to cut the tail. It reduces the amount of bombs it can drop by like half, and make everything besides the head relatively safe to stand(there's still some bomb pods at the base of the tail, but not many. The 2nd thing is don't bother going in when it's glowing. If your weapon has good reach, you can poke him from afar, but when he's glowing, his bombs detonate way too fast so if one drops at your feet while you hit him, there's a good chance you're getting hit, which can lead to death on tempered(or instant death if it's more than one).
Pokes on the tail is how you cut it, I don't know much about the switch axe but iirc it has some relatively fast long reach single moves, it isn't slow as shit like the greatsword. As IG, I use the forward+triangle attack which is just a small thrust, doesn't do big damage, but it has long reach and is very quick, so I can poke the tail then roll the fuck away like a pussy. Repeat everytime he does a stupid charge or whatever, just one hit, roll away then run like a pussy. When he's not glowing, can go a little bit more ham, but still no long animation shit. Also vitality mantle before you go in is good. Affinity booster when you get a chance to do damage(like if he gets fucked by a Rathalos or whatever else is in the area). I think fire mantle might be good too but not 100% sure if it works on the explosions or just the burning damage.
Once the tail is cut, you can kinda wail on the legs relatively safely, and Bazel has terrible defenses and basically no heavy armored place, so even though legs aren't a weakpoint, he'll still take a lot of damage. When he goes down from a KO or whatever, I still prefer attacking where the tail was which is still a weak point, than go towards the head in case bombs drop around his face.
I'm sure there's better, faster ways to deal with tempered bazel(other than just playing a ranged weapon obviously) but I'll take my 20mins hunt over carting 15secs into the fight because I hit it too hard and it dropped 12bombs mid animation.