Im not sure what your referring to with the polar bear here. Polar bears feeding effect is to lower the tourney cool down timer; At lvl 10 it lowers the timer 20%. That doesn't effect manual or auto fighting at all.
I can however at least provide some insight on the use or brown bear for tourney/spire.
First off..
30 provs on manual? My god... that must take an eternity to do. - While i can do manual, i fight with auto fight. Gotten pretty good at it. But even that can take some time with two cities.
As for the brown bear.. I personally fight 13 provenances at 5 star every week and the full spire. No 'resting weeks'. And iv been able to maintain this. Granted.. i went to lengths to try and make sure this would be the case. And while im still short of my initial goal, i found some unexpected boosts to help make up for it fully.
A few notes on my set up.
I always craft a pet food in the MA when it shows up. As well as any 50% building (magnificent mage multiplies, enhanced light ranged, unleashed unit upgrade. Note the unleash unit upgrade (health) is only +25%)
I have a lvl 10 fire phoenix for 50%+ strength boost, and a lvl 10 brown bear to help replenish troops.
AW:
Needles to increase training speed.
Bulwark and shrooms to help increase my training size. (4 armories as well)
Recently i placed a Monistary (MM) for boost in health, but thats still fairly new for me.
Non-Direct AW:
Prosperity towers, to help with supplies to train all these troops.
Golden abyss and mountain halls. Everyone and everything needs population. (Technically, the MH goods boost is nice to if you ever have to negotiate.)
Obviously, the higher level each one is, the more effective it will be.
Each person has a different idea of when to place the three +50% buildings as well as when to use a pet food.
I personally, will place the buildings at the start of tourney (not the start of spire). This means i get the full effect of them for the whole tourney, and the spire. With time instants you will win in the spire, you can power through any locked gates on the last day that you didnt open naturally.
Right now, im going through the mass of buildings i built up over the years. So iv been using 1 mage, 1 light ranged, and two 25% health's.
Note; one building reward in the spire is Dwarven Armorer. It gives +50% health (equal to two unleashed unit upgrades. So sometimes ill use one of them instead.)
For tourney i will use a pet food in the afternoon of the first day it starts. This allows me to fight the first round. And when i get up the next day, the 16h timer will hopefully be over, but the 24h fire phoenix pet food will still be active. So i can do a second round on the same pet food.
Since im doing 5 stars on tourney i can make one of two main choices. Either use 3 petfoods a week to cover the whole tourney period. OR i can do the first round with only the buildings 50% boost. Since its round 1, they should be a lot easier then later on. But you said you were doing up to #30, so im guessing they still get hard even at 1 star. But that leaves only 4 rounds left so i can only use 2 pet foods.
Dont worry.. im getting to the brown bear...
So thats what iv been doing the last several months. 13 provenances at 5 star along with full spire each and every single week; no break.
I start with my barracks squads at ~200 for each unit. Right now.. my over all losses are a little bit more then my unit creation. But i can still go a good month or more before my squad numbers become low enough to cause concern. (Im training around the clock 24/7. Thats why training size and speed is so important).
During that time, you will get a lot of time instant rewards in the spire.
So after that month or more when my numbers are low, ill feed my brown bear for the +50% unit production bonus. And i will fill up my barracks que, and start using those time instants. Ill have my whole army back up to ~200 squads all at once. And only at the cost of a single pet food. The key there is to do it in bulk. Not a few troops every day or every week. Get it all done at once to save your petfoods.
The only thing you have to take note about doing that... is supplies. Thats a lot of troops your training at once. So you will need a lot of supplies at once. So save up those supply instant rewards from spire as well. Or if you see a 100%, 50%, 33% supply instant in the MA, then craft it. they will come in handy. And, all that tourney and spire will mean youll have plenty of spell frags to craft with and relics for the catalysts. So its a nice big circle of goodness.
But doing this means i can do all my tourney and spire every week with no resting. And im only at the end of orcs chapter. (First tech of wood elves unlocked and then is topped for a while.) And in the end, dispite all the costs of doing all this, i still make/get more time instants and supply instants then i use. My cost of goods is very low, so i net out on that. I get more relics, spell frags then i use. Once you get to the point of being able to do this, im coming out with more of nearly everything in the end. Which feels weird... but i like it lol.
And i do all this with auto fight. So if your able to do better on manual then you might have less troop losses.
But the polar bear only reduces the cool down timer 20% at lvl 10 (from 16h to 12h 48m). Which is nice, dont get me wrong. But for what im currently doing, the brown bear came first.
Tho the timewarp AW (late in the game) at max level can reduce the cool down timer 80%. I would love to see having both the timewarp and the polar bear capped
lol