A sermon I heard, the 8 Dec 2019 installment of the Jesus Revealed sermon series, explained the background on why King Azaz rejected God's offer of a sign, as a background on another consequence of Emmanuel (because this is a Christmas sermon).
The historical background was that Syria & Israel formed a coalition to repel a much bigger "bully" Assyria. But because Judah (ruled by King Ahaz at that time) refused to join the coalition, they threatened Judah. But God sent Isaiah to calm Ahaz (Isaiah 7:1-9), and offered Ahaz to ask for any sign to authenticate that the prophecy is really coming from God. It was a very generous offer:
"Make it as difficult as you want—as high as heaven or as deep as the place of the dead." (Isa 7:11b).
In other words, God wanted Ahaz to see His greatness and wanted Ahaz to trust God for protecting His beloved nation. Because Ahaz was the rightful king of Judah at the time, it made sense God made the offer to Ahaz personally. As we read in the history of Israel & Judah in the Old Testament, the general theme was that God wanted the Israel / Judah kings to trust Him instead of relying on political alliances or military technology (that's why God said no chariots) and God has delivered victories against odds in his dealings with Ahaz's predecessors. Therefore, Ahaz could have rationally relied on God for protection against Israel & Syria, and again, a promise for protection was especially delivered via Isaiah on this very occasion.
BUT King Ahaz relied on bribing Assyria instead of trusting God 2 Chron 28:16-25. Basically, he was saying to himself: "No thanks, I trust my own arrangement for security." But of course he couldn't say that to Isaiah to his face, so he used a pious excuse by quoting Deut 6:16. That's why God was upset at Ahaz (Ahaz's human scheme didn't turn out well), especially because this bribery of Assyria led to the worship of Assyrian idols! If Ahaz would simply trust God's word of safety delivered through his official prophet Isaiah, he wouldn't have to bribe Assyria.
Therefore, the reason for Ahaz's refusal of the sign is not trusting God's word of safety, masking it with a false pious excuse of not wanting to test the LORD.