Your random wire tuning problem got me thinking that a QRP (i.e., small) three-way switchable UNUN (direct-connect, 4:1, & 9:1) would be a handy field operations accessory.

Try connecting the 41 ft random wire directly to the KH1. With a 13ft counterpoise, EZNEC modeling shows a resistive value close to 50 ohms and an inductive reactance close to 2500 ohms on 40meters. On 30 meters, the resistive value is about 450 ohms with inductive reactance close to 5000 ohms. So, your 9:1 UNUN should be helpful for matching on 30 meters, but not on 40. (Edit: The reactance values are bogus. I had added inductive loading to the model and forgot to remove it for reading unloaded reactive values. Here are the unloaded reactive values: 40 meters ^capacitive^ 550 ohms, 30 meters ^inductive^ 750 ohms. Of course, the resistive values were not affected.)

From the manual. The miniature ATU module in the KH1 has fewer L and C components than in the Elecraft KX2 and KX3 ATUs. This means it will not always find a very low SWR. The internal tuner is not as wide ranging as KX2 (due to small size of radio). Hope the following helps: If SWR is very high (> 3:1) after matching with the ATU, consider making length or other adjustments to the antenna or counterpoise. Tap ATU again afterward.