Some pics and a diagram would always help describe this better
From what I gather the dormer is above the bathroom on the second floor. How tall are the exterior walls you are placing the tub next to? Inches to the roof? Is this an attic remodel? The dormer has fixed glass or what type of window? Is there a bathroom fan? If not install a well insulated one w/a flapper for the size of the room and use it.
Lime would not be an issue here since other than having to keep it wet a couple times a day 2-4 weeks, and in a functioning bathroom that is easy. It is labor intensive but since you already have the lath an option. Lath is not cheap. If you are not familiar with lime get on the American Lime Associations website and read their publications. It is an acquired skill not a quick DIY.
Your roof is 2 x 6 lumber, the cavity
should measure 5 1/2 ". You could use denim insulation it dries fast, has borax or boric acid as an anti
fungi and fire retardant. I'd recommend mineral wool in a bathroom. It will provide R-21, in Detroit roofs s/b R-38+. If you have the head height add Roxuls (1-2)stagger 1.5" boards for another 3" R-6-12 to the ceiling I'm guessing is cathedral/vaulted? There is no flashing req'd on the underside of roof sheathing other than if you install a fan follow the mfg and silicone caulk any existing fan.
Walls should be insulated to code min there is R-20, if not Jami is correct they will thermally bridge into bath tiles, use fiberglass surround unit if you can. Tiles use a hardiboard cement board rather than Georgia Pacific's backer board with acrylics. You can also use the plastic you take off the walls to thermally break the wall studs and/or add some Roxul IS boards.
https://energycode.pnl.gov/EnergyCodeReqs/index.jsp?state=Michigan
I recommend removing the vinyl tiles over plaster. For quick, known, DIY, etc.... and easily accessible I'd use a fiberglass faced and taped gypsum board like GP DenseArmorPlus:
http://www.buildgp.com/densarmorplus-high-performance-gypsum-panels No microbial producing paper, FG just like mag board. I'd steer clear of DragonBoard or any Magboard from China too many quality control issues and little tech data to prove it's
sustainability especially in wet environments. Foreverboard here in NY is an option and the only US mfg if you fully understand mag board and the industry. Not DIY friendly. Better to go with a higher grade gypsum w/FG like GPs, etc. Due to expansion and contraction use it everywhere, all walls. It has a perm rating greater than 10, can provide some humidity buffering up to 5% it's weight....not as great as lime or clay but not bad. To get more add a mud hot coat using 1:1:2 (mud: type s lime: sand (80 grit jagged edge mason: mud: (Wespac, Murphy). Use a high perm siloxane sealer, lime wash or mineral silicate paint like ROMA or Heim.
I've been to this facility in Chicago before they are doing some interesting things with hemp/lime not far from you:
http://www.americanlimetechnology.com/