Solid Hardwood, Engineered and Laminate Flooring - Solid Wood to Tile Transition Height

I am preparing to put in a tile floor in the kitchen and a wood floor on the remaining area. The tile is 3/8" Porcelain and I plan to use a 1/2" concrete backer board. I will pull up the linoleum and underlayment. The sub floor is Chip 3/4" and is new construction (6 months old). The wood is strand carbonized Bamboo 5/8" thick. My question is this. The tile section will be 7/8" thick while the Bamboo will be 5/8" thick. I am going to have an arc so I would like to not use a transition piece and only a silicone joint. Any suggestions on how to get the levels correct? Should I put a 1/4" plywood over the subfloor on the wood section? I am a little skeptical but what about 1/4" concrete backer board on the tile? Any suggestions would be appreciated.
Thanks

Adding 1/4" under the bamboo would be the best method. The rule for tile installation is a minimum of 1 1/4" total subfloor thickness (3/4 osb + 1/2" backer = 1 1/4"), so only using 1/4" backerboard would leave you shy of that mark and may cause the floor to flex under the weight of the tile and mortar, leading to cracks, loose tile, etc.

Thanks tn.floorguy for the advice. I was coming to that conclusion myself but just needed that prod to push me in the proper direction.