If you can't turn off the Accompaniment while recording (and I'm not sure whether you're supposed to be able to do that on the DGX-660), there are a few things you could try.
(1) Yamaha keyboards recognize many different types of chords for controlling the Auto Accompaniment, and one of them is called "Cancel," also called "NC" or "No Chord" in some documents, for when you want to turn off the chords in the Accompaniment-- such as for playing measures where the section showing the guitar chords or other instruments' parts are marked "Tacet" on the sheet music.
To trigger this "Cancel" chord type, hold down three consecutive keys in the Accompaniment section of the keyboard-- for instance, C, C# (or Db), and D. Note that this key combination might trigger a minor 7th chord instead, so you might need to change the chord fingering method (but don't use the full keyboard method). The documents where this "Cancel" chord type is mentioned say that the keys which trigger it are 1-2b-2, or the root, flatted second, and second. So if playing three consecutive keys just triggers a minor 7th, try shifting one of the keys to a different octave, like C#1-B1-C2-- that is, play C# with your left little finger while using your left thumb to play B and C at the same time in the octave above.
I'm not sure whether this will turn off everything except the rhythm, or if the bass will continue to play.
(2) Some Yamaha keyboards have a Style Assembly feature that lets you copy an existing style and change it, such as to remove all style parts except the rhythm. The DGX-660 does NOT have that feature, but there are free utility programs that let you create new style files from MIDI song files, or edit existing style files:
My Software - The Unofficial YAMAHA Keyboard Resource Site
www.jososoft.dk
Explore essential utility programs for Yamaha keyboards to manage style, MIDI, and registration files. Enhance file functionality with PC-based tools and downloads.
www.psrtutorial.com
If you're trying to record with an external style file, you could just make a copy of that file on your computer (so you don't "ruin" the original file), load the copy into one of these utility programs, and delete all of the style parts except for the two rhythm parts (Rhythm1 and Rhythm2). Then you could use the new rhythm-only style while recording your song.
If the style you're trying to record with is an internal style, you can play it normally (just the Accompaniment, not playing along with it) and record it to a MIDI song file. Then you can either convert the song file to a style file and remove the unwanted parts (channels) from the style file, or remove the unwanted parts from the song file before converting it to a style file (i.e., same result, just doing the steps in a different order depending on which seems easier to you).
(3) You could also just record your song-- to MIDI, not audio-- using the full style playback, then transfer the MIDI song recording to your computer, remove the unwanted style parts (channels), transfer the modified file back to your keyboard, and convert it to an audio recording (if your goal is to have an audio recording of the song).
By the way, the two rhythm parts are channels 9 and 10, so in this case the "unwanted" style parts would be on channels 11 through 16.
(4) The "last resort" solution would be to use a DAW on your computer to do the recording. This would actually be the preferred solution for some people, because a DAW will give you much more control over the recording and editing process, as well as many more possibilities such as combining the tracks from your keyboard with tracks recorded using other instruments (guitars, vocals, etc.) or tracks that use sounds generated by virtual instrument plugins (which might sound better than the voices your keyboard has). But I realize DAWs aren't for everyone, so that's why I'm calling it the "last resort" solution.