Powernumpty

"I'm wondering if the kiwi will be able to handle it" I'm pretty sure downconverting that range has a lot of other VHF (+FM broadcast) mixing stuff to reject so filtering is important before and after the conversion, The Kiwi is just excellent test kit showing up any failings and mixing products.
Admittedly I was using one set up for 2m so birdies in the airband may have been intentional to keep them out of the required band.

Does make me yearn for a version that does 32MHz of VHF/UHF or even a modular one, lots of rf boards are being sold for self assembly RF chains.

The LZ1AQ amp will work as a simple dipole, I think the balanced antenna may be easier to get working and I'm a real fan of CAT5/6 shieled feeds for RX antennas now.
Mine is two loops with extra vertical sections and it is more sensitive in the vertical dipole mode but also more prone to the local noise.

I also have an Aerial-51 Model 404-UL, that I keep meaning to put up somewhere (when I win the lottery) virtually invisible wire, they do heavier versions that cover 80m. The thin wire on the UL is probably not suited for permanent installation. I'm not sure if your wish to avoid the wire is visual impact or something else?

You need a nice old linear to feed a mini whip. Check it gives the required voltage at the right load, some 12V will be 18V off-load.

There will be pick up on the feedline, it needs to be halted and/or fed to ground at the most affective spot(s), that is where the fun starts.
At the very least see the outside of the coax as a likely source of trouble from common mode signals, break it up (in common mode terms) and ground it out.

I tried my old 144MHz-28MHz transverter and was OK on some parts of the airband, not very clean though.
Perhaps you could send an enquiry to transverters-store.com, they seem good value even though you would be using just the RX side.
I looked for other downconverters and maybe due to RTL SDR @£10 world there is very little call for it so prices reflect the specialised nature.

You could try a reflector behind it. I don't think I'd get much here facing west but there could be some other interference desensing it.
I'd also carefully check the voltage at the SMA pins 3.3V.

Is there any way you could actually put the puck outside (away from other kit) even if you have to remove the HF antenna?
To me it would be good to test it where it should work before assuming there is some fault.