You can buy very accurate scales. A more portable one is going to compromise the precision but it is going to be able to resolve up to 0.001 g +/- whatever. I don't know how much of a difference temperature & humidity can make but you can create a microclimate if that's a big concern. As far as air buoyancy, you can calculate that and take that into account or you can build a small vacuum chamber.
Well, indeed - so get out there and do it, if you don't find the multitude of scientific papers, that show a wide range of experiments that all align despite using different methods, convincing. One point of note - poor terminology on my part - I said 'scales / balance' earlier. That's not right - you definitely want scales, and not a balance, as a balance will compensate for gravity error. You want scales, and you want to make sure you don't calibrate them after moving - clearly using a reference mass for calibration will remove the tiny error that you are deliberately trying to measure. You'd also need to make sure the the scales were perfectly level at each location.
I don't think you'd need to go to the effort of a vacuum chamber - air pressure does not change that much, as long as you are at roughly the same elevation, and you would need a very large change to see much of a buoyancy difference.
But ultimately what you are describing is essentially the Kern gnome experiment - admittedly a bit of scientific fun (as the real science has moved way, way past this level) - albeit changed to an altitude perspective rather than a latitude.
Sorry, but were you saying that if the gyroscope tells you you're flying level, that means that you're flying along a curved path because you're assuming the Earth to be a globe and level means curved? If so, I have nothing else to say to you to put it mildly.
Aviation gyro behaviour is very compelling evidence for a round earth. I'm not sure you've understood what is being discussed here. A traditional mechanical gyro will hold its attitude regardless of what goes on around it - that is why there are so useful in aviation, for both attitude referencing and other functions like turn indicators (a 'rate gyro') and heading referencing. Taking the attitude example, we take a full freedom gyro, attach it to a system of indication and use it to tell the pilot which way is up. This is great, but it is subject to a number of errors. Transport errors, for example, are where you take a gyro that is indicating level and move it to some other place on the earth. It will, without correction, retain its original orientation, meaning it will progressively indicate pitch or bank errors. The solution to this is to have a system that exploits the fact that pilots generally either use bank angles above 5 degrees, or will try to not bank at all - very small bank angles are unusual. So gyro systems are designed to have a function whereby at small (ie <5 degrees) bank angles, the system is assumed to be level, and is subject to a small gravity correction, which means it retains level as it moves around the globe. This is inhibited above 5 degrees, providing precision during manoeuvring.
There is a similar issue with heading referencing systems, which are subject to errors caused by the earth's rotation. Simple systems have a 'drift nut', which is preset during maintenance to the aircraft's local latitude - this provides a correcting precession to the system, counteracting the earth's rotation. On more advanced systems, there is usually a pilot-controlled dial to set the local latitude.
All of these errors are also present in ring-laser 'gyro' systems - these aren't really gyros at all, but rather sense rotation in all three axes - and similar compensations have to be applied for transport topple and drift as well as rotation errors.