My real concern is that (politics aside; this is not meant to be a political debate) there is no denying that many of our foreign trading partners have a negative view of Trump (and likely Clinton in some cases) and will be hesitant or outright refuse to do business with him.

This is a poor understanding of how business works.

It must be understood that refusal to do trade is a double edged sword. It will hurt the United States and it will hurt the other party as well. So by holding on to pride and refusing to negotiate with the Trump administration, the foreign trading partners may do significant damage to their own economies. Most of these foreign leaders are expected by their constituents to be rational actors and would be held accountable for their actions.

The next thing we must assess is the United States' relation with the trading partner. Do they benefit more from trading with us or do they benefit more from us trading with them? The United States is a large economy and a net importer. (Consider Cuba's stance on trading with the US.) Given this fact, the Trump administration is likely to be able to leverage the US' position to get better than a "free trade" deal.

Finally, in business, individuals just almost always negotiate no matter how absurd the other party's demand. Consider the most extreme case, the default: "We're not paying your money back." Even in this situation, debtors run back to the negotiating table to draw out how much they're willing to accept for the debt.

Even if Trump's demands are absolutely absurd, it is the nature of business to counteroffer him.