Schönberg conducted several performances of Gurre-lieder himself, and at one point indicated the need for a "fourth" mens choir, simply constituted of first tenors (possibly with second tenors and basses as well) to carry the Hauptstimme of the vocal parts. In practice, 95% of the time the Hauptstimme corresponds to the first tenors of Choir I, hence my recommendation that Choir I be slightly larger than either of Choirs II and III.
For the sake of completeness, here is the list of instances where the Hauptstimme appears against various voices:
Page 3, Bars 169 - 171: the leading voice jumps from Bass 2, to Bass 1, to Tenor 2, to Tenor 1. (The three choirs are in unison.)
Pages 4 - 6, Bars 172 - 188, second beat: Choir I, Tenor 1. On the last crotchet of bar 188, the Tenor 1 of Choir II takes over the Hauptstimme.
Page 7, Bar 192, last crotchet: the Tenor 1 of Choir I takes the Hauptstimme back from Choir II, and continues for the next 70 bars.
Bars 261 - 263, last crotchet: the Hauptstimme follows Choir III Tenor 1. Bars 264 - 266, third crotchet: Choir I, Tenor 1. Bars 266, last crotchet - 270: Choir II, Tenor 1. Bars 271 - 276: Choir I, Tenor 1.
Bars 279 - 283, the Hauptstimme follows Choir III Tenor 1/Bass 1/Bass 2.