Given that all flat earthers seem to reject the existence of gravity, I would like to know what force FET claims is responsible for the Sun's movement?
Not me.
Let us see what force I. Newton claimed was/is responsible for the Sun's movement.
Newton believed that there are TWO GRAVITATIONAL FORCES AT WORK:
1. Terrestrial gravity
2. Planetary/stellar gravity
Here is a letter from Newton to Halley, describing how he had independently arrived at the inverse square law using his aether hypothesis, to which he refers as the 'descending spirit':
....Now if this spirit descends from above with uniform velocity, its density and consequently its force will be reciprocally proportional to the square of its distance from the centre. But if it descended with accelerated motion, its density will everywhere diminish as much as the velocity increases, and so its force (according to the hypothesis) will be the same as before, that is still reciprocally as the square of its distance from the centre'
A clear description of PRESSURE GRAVITY.
Newton still thought that the planets and Sun were kept apart by 'some secret principle of unsociableness in the ethers of their vortices,' and that gravity was due to a circulating ether.
Isaac Newton speculated that gravity was caused by a flow of ether, or space, into celestial bodies. He discussed this theory in letters to Oldenburg, Halley, and Boyle.
Newton, student notes on Descartes:
Gravity is a force in a body impelling it to descend. Here, however, by descent is not only meant a motion towards the centre of the Earth but also towards any part or region...
His belief at that time was that, to quote Westfall, ‘gravity (heaviness) is caused by the descent of a subtle invisible matter which strikes all bodies and carries them down'.