Installing a Heavy Sliding Barn Door into Metal Studs

I’m looking to install a single, sliding barn door in the 2nd bedroom of my condo. The door is solid MDF and weighs 200 lbs, including the metal rail and sliding wheels. And being in a condo, I’m unfortunately plagued with metal studs.

To date, I’ve installed a 1x4x80 wood backing board across 5 metal studs, with strap toggles going through the studs. The rail is 78.75″ and the door is 96×36″. I’m also planning on driving two additional screws per stud, on top of and below each strap toggle, for additional support. My plan is to install the rail onto the wood backing, with the provided 4″ lag bolts going through the wood/drywall.

Two questions:

1. Will my planned installation support 200lbs? Or should I consider cutting out some drywall and adding horizontal wood blocking in-between the metal studs?
2. At least one (maybe two) of the pre-drilled rail holes align with metal studs. Will I have a problem getting those to go through the metal studs behind the wood? Any best techniques for doing so?

I’m less worried about the board pulling out of the wall, and more worried about the rail pulling out of the wood. In considering how much of the 4″ lag bolt will be exposed behind the drywall, here’s the thicknesses of everything in-between…

1/4" Rail --> 1.5" Spacer --> 3/4" Wood --> 1/2" Drywall 

So, add that all up and there will be 1″ of the 4″ lag bolt exposed behind the drywall. Will it really make that much of a difference if that 1″ were to go into wood blocking?

I’d very much appreciate any advice! Pictures below:

1x4x80 Wood Backing Installation:

4″ Lag bolt with spacer (and also a lag bolt with a lag anchor, presumably for drywall only installation)