DOT baggage rules should apply as the ticket touches the USA. The allowance of the first marketing carrier (UA) between the first two ticketed points (LAX and KMQ) should apply to the entire ticket. Mixed cabin itineraries are a mess, but I believe the best reading should be that your allowance is 2x70 because the underlying coupon is Business. However, if you need the overweight bag I would be prepared for an argument at the counter in LAX. Every other counter should be giving you 2x70 or their own J allowance, whichever is higher.

You will give your bags to UA at LAX, and I think it is unlikely they will short-check to TPE although you can certainly try.