Never needed that much time at IAH, elite or otherwise. 90 minutes is what I recommend.

You can do online check-in 24 hours before departure. Even for internatiional flights.

Drop you bags at Terminal E. There may be a line, but usually not too bad at around 7-7:30am, except on like Saturday morning on a long weekend.

Security line at E is also usually short. Your morning Express flight to ZIH will depart from either Terminal B or D. For D, just walk over. For B, take the TerminaLink, also fast.

Or if the security line at E is super super long, then take the underground train over to B (or go underground and walk to D). And reclear security at either of those terminal. There are so multiple check points at IAH, so there should be one that's not long.