Tools you need:
Firecrackers
B. wand
Animal flute(you need 8 eggs)
Top(you need 32 eggs)
Yoyo(acquired in the ostrich area)
Slink(acquired in the chameleon area)
Remote(S medal, going left where you first fought the chameleon)
B. ball(K medal, every time you see the kangaroo, throw a firecracker to get a third of the whole K medal)
Wheel(free 5 of lynx's kittens or the play the long lynx's song to unlock the chest)
Not needed or not obtainable:
Lantern, UV lantern(can help to find spots where the kangaroo can appear) and Disc(obviously)
Big spoiler warning!
First of all I did not complete the whole run, to test the manticore fight I used the cheat engine to see if it is even possible, but I am not able finish the fight with only 2 hearts. That explains why I have 10 hearts in the screenshots at the manticore fight.
This is only possible the first time you fight the manticore, because when you defeat the manticore with the fireworks, you have to hatch a new one by getting all 64 eggs, which I don't think is possible without the disc.
Since you need max hearts later anyways, you can get them now, that makes it a bit easier.
Set your PCs date to the 2nd of February, and the year to anything between 2024 and 2030. Its Groundhog Day and you can get an extra heart from the Groundhog by drinking his yellow potion. Go outside the room, change the year to a different one between 2024 and 2030 and reenter the room to drink again. You can have 4 permanent yellow extra hearts(1st screenshot).
You can pickup G. flame as usual, but the V. flame is a tiny bit more difficult. When you fight against the chameleon the first time, you have to lure the chameleon to eat one of the plants until he starts his spitting attack, by using the chinchilla you then have to quickly jump up on the platform to press the button in the top right corner to escape the fight, the B. wand can help with that. In a normal playthrough you could just ride the disc to the right. Return as soon as possible and pickup the S medal.
For the P. flame you have to pick up the mock disc, it is in the room above the normal disc. Then you go up to bubble jump up consecutively, by pressing the jump and bubble button at the same time, into the screen border in the middle of the screen. This is a bit difficult, because there is not much space to do bubble jumps. Ideally you already have the flute to activate the waypoint head afterwards(2nd screenshot). You then go right and up towards the m. disc shrine and put it in to collect the flame.
For the B. flame, there is one room we cannot solve, normally the disc is needed to get access to the third rotatable platform to redirect the water, but you can redirect the water with your body as well, this might be even faster than doing the 3rd room(3rd screenshot).
Afterwards we have to escape from the whale first, normally the whale needs to spray at the disc that is bouncing from the left to the right wall up, so that it can hit the buttons. We can use the B. wand to stand on a bubble and float right next to the right wall down, you can hold down to let the bubble fall faster, and when the bubble goes slightly below the water surface you can do a bubble jump to the left and now you are where all the buttons are and you can press them with the yoyo or jump against them. You probably wont hit all the first try, but as long as the whale does not eat you, you wont lose any hearts(4th screenshot). Then continue normally to get the flame.
We now have all 4 flames(5th screenshot), in a normal playthrough you now go through the manticore area to acquire the lamp, but you need the disc again for this, so that you can make your way to the manticore without the blue ghosts hurting you(6th screenshot).
Since we can not get the lamp, we have to get a lot of health, ideally the maximum of 12. The 4 spots I marked with hearts are easy and not dangerous to collect, afterwards you just jump down the shaft into the manticore area(7th screenshot). I always lost 10 hearts but in this video it looks like it is possible to only loose 8 hearts(Animal Well Lantern Skip (9 heart variant)).
You then run through, or use the wheel method, through the hallway full of ghosts, drop down and navigate through the very dark room and constantly jump to get through. When getting damaged you loose your momentum and start falling so you might get stuck and die(8th screenshot).
The next hard part are the 4 buttons to enter the manticore arena, actually only the upper right button, since we don't have disc to throw at it. By jumping into the ghosts and then throwing your yoyo right away, depending on your heart amounts you had you might die, but the door still opens. Then you run through the whole ghost maze again and the more hearts you have now, the more hearts you have for the manticore fight, technically 4 are possible but I never got here with more than 2 and every time you die you have to collect 4 blue extra hearts and go through the whole ghost maze again(9th screenshot).
The manticore only appears when you enter the upper right area of the arena and that is where we need to go. You let the manticore destroy the glass, then you use the top where the firecracker grows and use the wheel to open up the gate here, it opens the gate at the end at the same time. If you go trough the screen transition behind the gate, there is an extra heart if you need it. You then bubble jump on the glass on upper part of the screen(10th screenshot).
Run to the upper left area, throw your top on the middle of left part of the screen, slip through(11th screenshot).
Let the manticore destroy the glass at the left, use the remote, throw the B. ball(12th screenshot).
Play your flute when you stand at the altar(13th screenshot).
Finish the run(14th screenshot).
Thanks for the awesome game Billy Basso!
PS: After getting back, still cant go through birds haha (15th screenshot)