Plating 96-well bacterial transformations
I think your approach very much depends on how essential it is that you get discrete single colonies, and if this is a one off or a regular process. To preface, similar to what Maximilian Press said, using a single welled plate like a nunc omnitray would be your best bet.
The main consideration you need to take into account for high-throughput transformations is the transformation efficiency (TrE). Naturally in a big petri dish, 100 µL of a single 1:10 or 1:100 dilution of your recovery culture can bring out nice discrete colonies. When you start reducing surface area and volumes, the chance of discrete colonies obviously diminish.
The best way I've found is to do a test transformation using the backbone of your choice and gauge the average TrE over a few runs to hone in on a 'one size fits all' dilution/volume, then apply that to your full high-throughput run. Assuming your inserts are of similar size, you should see a relatively similar TrE between them (bar edge cases). Another option would be set up three 96-well plates and do three separate dilutions, 1:10, 1:100 and 1:200 at 100 µL a pop so you at least get some fudge factor.
In terms of the chemical transformation itself, doing them in a 96 well plate format will guarantee lower TrE simply due to the characteristics of the plate itself. Thicker plastic and sub-optimal heating (if you aren't using a thermocycler and PCR plate) requires a slightly longer heat shock, and recovery in media requires much more vigorous shaking due to the mixing characteristics within the wells.
If you don't care about plating out in 96 well plates, like Maximillian Press said, use a single welled plate and format it in a 96/48 well fashion. Will save a massive headache when pouring. Just make sure you leave it to dry to prevent any running culture.
In terms of actually growing the colonies in a 96 well plate, if you're making the agar yourself I'd recommend adding an extra 0.5% agarose to thicken it up (I doubt you'll be using one, but make sure your plate is non-tissue culture treated). Wait for it to cool just above its solidifying temperature (around 45 °C in my experience) and then transfer out using a pipette or low volume serological pipette to reduce the meniscus formation. If the agar forms a high angled meniscus, your colonies may form smears and will not be discrete. Leave the plate out to dry as well, colonies may also smear from the agar being moist.