The Lydian mode is the brightest-sounding mode and I am interested in how to utilize it.
Diatonic chords in Lydian = I, II, iii, ivø, V, vi, vii
What chords are tonic, subdominant, and dominant, in terms of functional harmony?
Functional harmony and functional concepts like dominant and subdominant do not apply to modes, just to major and minor. Modal harmony is a separate subject and boils down to emphasizing the tones that make that mode unique — #4 in the case of Lydian. So a II-I cadence would be more "Lydian" than a V-I cadence.
V>I would be convincing, if functional harmony is what's needed, sound-wise, and will be a deciding factor to keep a piece in Lydian rather than Ionian. But there is no IV chord - it's ♯IV, and that won't have the same functional harmony as the Ionian or perhaps even the Aeolian (which is debatable with no proper leading note). So, really, there's no direct comparison between I IV V in major, and any, for that matter, of the modes which emanate from that major key.
I suppose that a II6-V-I may sound good enough cadence-wise. The Major II works as a subdominant (or, in my opinion, more accurately, a pre-dominant) in a Major key and thus may be used to emphasize the tonic note if not the key-hood vs mode-hood twixt Major and Lydian. I think II6 may emphasize the sharp fourth scale step better than a root position chord (maybe even II65 would be better.)
I don't know much about "modern" mode harmony but I would guess that a Lydian mode needs to emphasize the tonic note and the notes that differentiate the major key on that tonic with the mode.
The subdominant scale degree is a perfect fourth above a tonic and is the fourth scale degree.
The lydian mode's fourth scale degree is an augmented fourth above the tonic.
There literally is not a subdominant in the lydian mode.
In functional harmony the terms "pre-dominant" and "subdominant" are often used interchangeably, but technically you can make a distinction between the two. For example, the
iii chord in a major key could be called a "pre-dominant" - by virtue of not being the tonic or dominant chords - but it clearly is not a "subdominant" chord.
Beyond those technical points, I think you must ask yourself what you want to do.
If you're goal is something like ancient modal music, you should understand a flat was often used on the augmented fourth degree in lydian modes, which ends up making the music sound very much like major key music.
If you want modal harmony, but with something of tonal harmony's solid tonic/dominant relationship, handle the fourth degree very carefully (many might say "avoid" it), melodically a major pentatonic scale will avoid the fourth scale degree, while providing a solid sense of the tonic, some careful use of the raised fourth as auxiliary to the dominant degree should not undermine the sense of tonic/dominant, try
vii (minor triad) as pre-dominant chords moving to
If you want modal harmony with an "ambiguous" sound, embrace the sound of the "raised" fourth and the sort of "anti-resolution" is creates by tendency to move up to the dominant instead of down to solfege
MI of the tonic chord.
If you go in a jazz direction, the raised fourth scale degree, and the lydian scale, are important aspects of modern jazz harmony. In this context "modern" is something like post swing era, the music of people like Thelonius Monk, John Coltrane, etc.
Getting back to functional harmony and tonic/subdominant/dominant, the other mode that stands out as not fitting into the functional harmony (or major/minor harmony) system is the locrian mode, which has not a perfect fifth but a diminished fifth above the tonic for the fifth scale degree.
Lydian and locrian are both at odds with conventional, functional harmony. The other modes - dorian, phrygian, mixolydian, aeolian, ionian - all have fourth and fifth scale degrees which are perfect fourths and fifths above the tonic. All have triads of major or minor quality on each of those scale degrees - along with their tonic chords - excepts the diminished chord on the dominant in phrygian, but phrygian is its own special topic to get into another time. The point is lydian and locrian are special because they introduce augmented and diminished qualities to the tonal scale degree relationships between
With modes dorian, phrygian, mixolydian, aeolian it is fairly easy to think of them as "modally flavored" alterations of major/minor tonic, subdominant, and dominant functional harmony. But with lydian and locrian you should recognize the destabilizing of the tonal scale degree relationships and the resulting departure from functional harmony.