You just nailed the fun aspect of Bear builds: running out of stamina.

To be effective with this build, you need to be super efficient with all of your moves to avoid wasting stamina. The basic punch is what you'll want to use for at least round 1 in all of your fights at the start. Uppercut you don't pull in until round 2 or 3 once you've got a good stamina lead (but be prepared to switch back if you get a string of bad luck).

Next, keep a close watch on how much energy potential upgrades will cost vs how much damage/accuracy they have. Crosspunch vs Boxing punch is a good comparison (we'll use the strength stat of 13 I have on my current save). Crosspunch deals 28 damage, has 80% accuracy, and uses 22 stamina. Boxing punch deals 22 damage, has 85% accuracy, and uses 10 stamina. The amount of extra damage you can deal with Crosspunch just doesn't compare to how efficient Boxing Punch is. For every 1 Crosspunch, I can easily get out 2 Boxing Punches and deal up to 44 damage.

Aside from picking skills that are energy efficient, don't overload yourself with skills during fights either. Just because you pick 3 attacks doesn't mean you'll attack more often. Using more attacks just means that you'll pick one of them at random when you attack. It's better to balance out attacks, defense, and debuffs. It's much better to use Boxing Punch, Triple Punch, and Block than to use Boxing Punch, Crosspunch, and Uppercut. Remember that skills like Exhauster will not cost you any energy during fights, so if you need to choose an attack modifier or an extra attack, go with the modifier.

It's a good idea to go through the entire tree and find which upgrades you plan on getting in advance. This way you don't get side tracked getting other upgrades and starving yourself later on skills. As for which stats to train, the most important ones are (of course) strength and stamina. While you can get away with completely ignoring agility, don't. Keeping your stamina at about 1/2 of your strength is a good idea since the game uses the levels of all 3 skills together to determine your max health (not sure how the calculation goes unfortunately). If you have 13 strength and 9 stamina, it's safe to keep agility around 6.

Finally, here are recommended upgrades:
Closeline - high damage, low stamina usage
Exhauster - drains more stamina from your opponent when active
Wreck'em - use less stamina when you miss
Diehard - gain extra stamina back after getting knocked down

Hopefully this helps. It's not 100% accurate, but it should give you a better idea of how to play a bear build. Good luck.

I beg to differ in view of this: https://steamhost.cn/steamcommunity_com/sharedfiles/filedetails/?id=635124748

Still 95% win chance, only 2 fights to go before the island, way of the turtle. This is the highest I have ever gotten in a game - I lost way more than 7 fights with tiger (while struggling towards the good dodges and such), and had 93% win ratio with Bear. So far this turns out to be way better.

Granted, Turtle requires some foreknowledge of the game, selecting a very precise amount of skills and probably breaking bad instead of living in a mansion. I am having a blast with it, and don't even know what to do with my skill points... Has to be said, once you get the last 4 skills in the turtle tree, you are unbeatable. Cutting in-combat stamina regen for both of you while leeching it from the opponent is crazy. High agility enemies with reflect are a minor pain because they don't really reflect all that much and I am looking at over 350 health now.

Yep, turtle is fun.