OK, before we talk about what you can add for more insulation, let's talk about why it was cold in the first place. One: No sheetrock behind the tub, just insulation. Without an air barrier the space below the tub would have gotten very cold and circulated air up into the studs above. Two: Shower curtains. Anything that blocks the heat, reduces the temperature. Three: The ceiling below is insulated. But how? The rim joist is the outside perimeter of that cavity. It is very common to have air leakage between the foundation and the sill plate and between any and all wood that has seams extending to the outside. If you can get back into that basement ceiling, you need to air seal that rim joist. Once sealed, you need to pack that cavity full of fiberglass under the entire bathtub area. Four: I've never done this, but I see no value for having the open space under the tub. It's a dead air space and from my point of view could be filled with insulation. How to get it in there, I don't know, but take a look. If you can get back into the basement ceiling, you might be able to cut some access holes and cover them when done. Now, your 3 1/2" insulation is probably R-11 or R-13. Your options are replacing it with; spray in foam, they have some DIY products; cut and fit rigid foam and fill the stud bays; or they make a high density 3 1/2" fiberglass R-15. But you will need to be sure you have sheetrock, or other, floor to ceiling to re-establish the air barrier. Adding a layer of polly wouldn't hurt. If you are considering insulating other areas of the house, where you could use cellulose, then maybe you could justify getting a machine and blowing it in. One more thought, I get involved with helping seniors create a warm space, where they can stay warm without heating the entire house to 80 degrees. If a toasty tub
is important to you, there are many ways to heat it up, especially if it is well insulated from the outside. Do you have hot air or hot water for heat? Your turn Bud