Hi Justin,
The idea of putting a lavalier mic directly into a recorder on each talent is a very good one. I wouldn’t recommend using the H2 however as it does not supply any power via the microphone port. You would therefore need to use a dynamic lavalier microphone and I don’t anticipate you will be happy with the sound produced by such a mic.
Additionally, you will be compensating for inevitable drift between the 4 individual units and camera since those devices do not have accurate enough clocks to prevent drift over the course of only a few minutes.
Most lavalier microphones have 2-wires, thus are not balanced. They can usually be ordered with “bare leads” or “stripped and tinned.” You may (or have a professional) solder any connector you want, in your instance, 1/8″. Some lavalier microphones have 3 wires but it is common practice to wire them more simply as two-wire mics. The lack of a balancing circuit (my interpretation of your term “grounded”) is not a major detriment. If you literally meant “grounded” as “connected to 6 feet below the earth,” I don’t anticipate this would reduce interference. Choose a lavalier mic with good shielding and keep cell phones (especially GSM) and walkies away.
Audio Technica is a reliable option in the price range you mentioned. If you can choose a recorder that provides electret power at a level compatible with a mic you like, you may find success.
The audio professionals solution of what you are attempting is the Zaxcom ZFR100 for $995 which is a bodypack style recorder with a professional lavalier microphone input. It records timecode stamped audio to a miniSD card and is a great choice for extreme sports situation or other situations where talent is too far from the sound mixer to reliably receive the a transmitted signal. They also built this recording function into their 900 series wireless systems which are truly revolutionary.
If you need this for a specific project you can rent the ZFR from a place like Coffey Sound. Otherwise, if you’re intent on purchasing I would suggest you consider the Zoom H4N and a hard-wired lavalier that accepts 48V phantom power via XLR. Otherwise you could opt for a Marantz PMD620 (which claims to provide mic power of some kind) and an electret lavalier mic.