OK, that's a bunch of questions so let's unpack them one at a time.
I prefer HF because it weighs less and runs cooler. That's just a preference. Both work.
cost of building a LF-Inverter was higher vs. the same wattage for a HF-Inverter;
I don't know who told you that but I doubt it's true. The LF one needs a huge transformer but that's just a chunk of iron with some copper wrapped around it. The HF needs a ton of engineering to get the sine wave right so it has double or triple the amount of logic circuits. It's like trying to figure out if it costs more for GM to make a front wheel car or a rear wheel car. They're just different.
the stability of running high loads (10A+) for long periods of time was safer using a LF-I vs a HF-I
Safer? That's a funny word because it can mean a lot of different things. If we consider the inverter to be well designed and built - a good industry name with some reputation like Aims, Magnum, Trace or Xantrex - then both LF and HF can run 10A+ all day long if the batteries hold out.
when it comes to starting heavy duty motors, like table saws or other 1000W+ loads, I don't know if the LF-I will work better than a HH-I if you just have 2x the rated wattage for each inverter being compared.
If they are well built both should work. Table saws start up at no-load so they are easy, they don't load up until the wood hits the already running saw blade. AC compressors are probably the hardest to start, at rest the pressure in an AC system equalizes itself so that the initial push onto a compressor can be substantial.
if I had 300aH of LiFeP04 batteries to power a 8A portable AC and residential refrigerator, all day long in my RV, would you buy a high quality $1,500 PSW, 1500W, LF-I ...or would you buy a $600 PSW, 2000W or 3000W HF-I?
Well can we assume the residential fridge is a newer type and it's current draw is 5A or below? That would make your power requirement 13A = 1500W. A 1500W inverter trying to run both of your units would be running at 100% capacity all day long, that's not good no matter if it's HF or LF. So with the double the power rule I would recommend a 3000W inverter. Since 3k inverters are harder to find I could also recommend a 2500W as being sufficient, you would be running it around 60% load and it should be fine to do that all day long.
If you can afford high quality buy this:
Inverter 101:
Remember it's the heat that destroys them not the surge. They are well designed to handle the surge and there are dedicated circuits to cut them out or clamp them down if it's too large. The power transistors that are used typically have a surge current that is four times the regular operating maximum so a transistor that can safely run 50A continuous can handle a pulse of 200A.
The big difference between HF and LF is where the energy used to make the sine wave is stored. In LF units a large transformer is used and very few capacitors so the storage bank is inductive and you're drawing from the magnetic flux. In HF inverters there is no large transformer but there are huge banks of capacitors and you draw from a capacitive energy pool.
HTH