Without knowing for sure your subfloor and underlayment are probably adequate. 1/2" Hardi backer board should be all you need to add.
The joists need beefing up for sure. Add one between each original and crossbrace.
Why a cast iron tub? Acrylic is lighter and warmer.
1/2" hardi and another 1/2" for thin-set, wires and tile.
Install hardi then tub, tile, toilet and vanity.

Thanks Don92 for the info on the joist. I just posted a question on the remodeling forum about how to add joist. I told my DH about the cross bracing suggestion you made. The rest of our house is braced that way. Our basement is not finished so easy to see. This bath is over a crawl space unfortunately and difficult to work with. I think I mentioned that this bath was an add on to the original house. The rest of the house is joist properly.

The reason for the cast iron is tradition & durability. I agree that acrylic would be warmer. We have something like that in our other bathroom and I don't like how the finish has lasted. Though it actually might be fiberglass & is very old too. I know that American Standard has a product that sounds like a cross between cast iron and a synthetic product. Its called Americast. I might consider that tub but need to see it first. It has an enameled finish I think.

Background info: We live on a family farm in the midwest and use the tub for more than just the occasional bath. We may fill chore buckets, bath dogs, etc. I don't think acrylic will hold up to that level of use. We are the 3rd generation to live here. The old cast iron tub that was subjected to this use has lasted since 1950. It is actually in ok shape, no rust. I had considered taking out the tub altogether and putting in a utility sink but it is nice to have a tub in this location for kid baths. My kids have fond memories of bathing in this room just off of the kitchen. Being off of the kitchen I want it to look nice for company too.

We anticipate at least another 40 years of service with this remodel. While aesthetic design is in play here so is longevity of fixtures.

Old midwest country family here too so I understand the need for the cast iron tub; say no more. Just make sure you support that floor correctly. It wouldn't take much more than a little elbow grease to just remove all the floor so that your work area is all exposed. You would be more inclined to do all the work properly.

Don, you're making me faint with the removing the floor prospect! But my husband says you're probably right;(. Though he does think that we can get in there and do this job without the tear out. He has done a lot of stuff over the years being a farmer, jack of all trades, and in a past life an aerospace engineer (the math!). We have a 20 year old son who can crawl! So that space will be accessible. But I am listening to your caution of "You would be more inclined to do all the work properly" if the floor was taken out and area exposed. I added my link to the joist info request on the Remodeling Forum.

Here is a link that might be useful: Adding Joist to Existing & Intact Floor

I have been a remodeler for 35 years and time is money. If it was just doing a spot of plumbing in the crawl space I would deal with it but adding joists is the deal breaker. 4 hours of tearing out the floor and 2 hours of plunking down 2 layers of 3/4" plywood would be well worth the time spent. Also it would give me an opportunity to shim and level the floor and end up with a much better result.

Something to consider Don, thank you so much for your help in this problem.

One of my hold backs is my fondness of that old wood that is under there, even if I can't see it. It's all in good shape and the floor as is seems very strong. When I jump up and down I only hear a faint short tiny squeak.

I am glad to be getting the issues discussed now because there is time to consider issues without being rushed. This job may not start till next spring, or never, it depends on the corn & bean prices!

Hey Don, I told my DH what you said and my response to you. His reply was: "now he knows he's dealing with a nut case;)...welcome to my world!"

humph

A "fondness" for subfloor? You must meet my wife. She has a "fondness" for an old tobacco barn on our property that you can see out the front window.I can't stand the thing! Your husband and I should be grateful our wives have a fondness for old beat-up things. Less chance we will be discarded for a newer model.
www.ivyway.com

Lovely stuff and the kids are cute! But where's the barn?

I'm going to brag now, here is my first tile job, individually laid. I cut the soapstone too and with the help of DH installed it in our small kitchen remodel. Other aspects of the remodel where done by others:

The barn is there! Now that the leaves are gone it is glaring.
Great job on the kitchen! And good luck with the project.

Thanks Don!

looks good