Hello!
I think you have a heck of a problem to deal with.
What most people do not understand is that there are two types of ground.
The ground everyone thinks of day to day, the `earth' ground in your house that all the grounds in all the outlets are tied to. Here in the states this ground it tied to the neutral at the main breaker/fuse box.
Other utilities servicing the house (cable, gas, telephone) all tie to this same ground.
The "OTHER" ground is an "RF" ground. Yes, it is in fact the same ground. However, with "special" qualifications. An RF ground needs to be able to pass current at an RF rate. Where the standard house grounds carry 50/60 Hz. The RF ground is operating at several hundred hertz to gigahertz.
Most house wiring fails miserable at passing RF frequencies (but do a reasonable job at transmitting them).
Over the years, I have been involved with locating/eliminating interference for short wave (back then who knew that short wave was really pretty long?)
HF (amateur radio) FM VHF/UHF (both amateur and business band) Television and some microwave. Some of these were at the transmission side, most on the receiving side.
I need to make the disclaimer that I do not have a "whole house" inverter or LED lights. However, I believe that RF interference regardless of the source is still RFI and needs to be addressed the same.
RF noise is "radiated" thru the air and along power wires. In addition, any attempt to fix the problems needs to address both.
To add to your problem you probably have more than one source of noise.
Your inverter (I am guessing) is putting out one type of noise (thru the power wires and radiated from the wires) and the LED lamps (each one that is tied to the inverters `grid'). The LED's most likely are adding their noise to that same grid. (Anyone remember BFOs or intermod?)
This mix of frequencies can mix with each other and create a completely different frequency of noises. Add to all this are the "other" noise sources in most households such as your computers, wireless modems, radio receivers, microwaves, cd players and on and on.
As the others have already stated, ground ground ground.
However, be very careful how and where you ground things.
I have seen cases where grounding actually made things worse.
Be aware of ground loops,
Do not be afraid of using too many ferrite beads on the wires feeding into and out of your inverter as well as each of the LED lamps. Make sure they are located as close to the inverter as possible (with in inches).
Cap type filters can be effective but are frequency sensitive and can be very difficult to match to an "unknown" frequency.
I'm "Guessing" that your inverter needs to be located as close to your grid/breaker box as possible, and the ground tie in should be made though a ground strap (a flat braided cable).
Once the noise has left the cables and becomes radiated the chances of fixing it at the receiver is just about zilch. At this point, it is all about Signal to Noise ratio. It is then too late to deal with the noise; all you can do is improve the Signal (a bigger better antenna and/or improved transmission cable between the antenna and receiver). YES, you should have your radio and antenna grounded, but all that will do is improve your radios reception of both the signal AND the noise (not touching the debate RE: lighting protection).
Good Luck!
ax7
Mark