First let me say that there is a difference between between constructive pain/fatigue and destructive pain/fatigue.
Whenever you do something new on the guitar, or use muscles in a different way, you’re going to feel some pain or fatigue. Muscles get stronger by breaking down and building up (everyone who lifts weights knows the drill).
At the same time too much pain is likely caused by improper hand position and too much tension. Or more accurately too much tension caused by improper positioning and trying to compensate with more force.
Now, here’s the rub… It’s almost impossible to get the optimal position when you first start learning. It takes practice and constant adjustment. Partly what you’re doing is paying attention to how it feels and making adjustments so you get a better sound with less effort. It does take time.
Here’s one thing I can recommend, but others might add some suggestions as well.
Practice the bar chord exercise a little bit each day consciously and slowly. Pay attention to how your left hand feels. When it gets tired stop and take a break. When you first start this you may only go for a little while. As time goes on your hand gets stronger and your positioning will get better so you low for longer periods of time.
The reason I recommend this is that this exercise narrows down what you’re working on. Even that 5-Minute Practice Session that you’re describing will force you to change positions several times.
You might even want to start the exercise on the third fret. That’s because the first fret is actually the hardest. Starting on the third fret will give you a chance to work the muscles without too much strain. Then, over several sessions, back down to the second fret… Then the first fret.
At the same time watch the videos where I play a bar chord and look at my left hand. Keep making adjustments as you notice yours looks different. Combine that with listening to the strings and trying to get a better sound. What I’m describing is going back and forth between a visual image and feeling the strings and neck of the guitar. I’d even recommend closing your eyes sometimes so you don’t see your fingers and just try to feel what works best. The two ideas work well together.