Must be that time of year when people are driving in the dark again.
Not quite sure what you were trying to get across here, but when I said "identical" I was referencing physical fitment on the vehicle. The HID projectors most certainly have different optics than the halogen projectors, which is why the OP can't just throw HID capsules in his original housings. Installing OE HID assemblies will solve any light output issues, as long as the wiring can be adapted (should it need to be at all).
As much as I bash drop in LEDs, many of the OE halogen projectors (including the Japanese ones) do really well with a rebased HID conversion kit in them. I have said HID conversions in two vehicles that are factory halogen projectors. The additional output brute forces rather than solves the problem of low efficiency (aged) housings where the lens and/or reflector (projector bowl) aren't passing all the lumens to the road.
Those gains, are however generally temporary because the additional light/heat of the HID (including poorly controlled UV output of cheap conversions) makes quick work of whatever reflective coating is left on the projector bowls.
I personally stick to 4k-5k bulbs. The "innovited" kits are pretty good for the cost but there's a definite improvement to Morimoto branded rebased bulbs, ballasts, ignitors, etc. Reiterate, please do not put these kits in a fog light (no glare shield), high beam (they don't like to be cycled), or reflector-based low beam.
For the OP, it might be well possible to bolt in a set of the factory (or aftermarket replacement) HID housings, new or in good shape. For highs, likely 9005, convert to 9011 Toshibas. For lows, rather than use the OE D4s bulb, use a D2s with aftermarket ballasts and ignitors and plug into factory LB supply wiring. Morimoto bulbs and ballasts can be had for under $100. The HID bulbs would be $50 to $120 depending on taste. High beam bulbs, $45. Housings, whatever. The projectors themselves wouldn't be state of the art, but nonetheless pretty close.