What dye to use to give you the results you want with the garment you have?
Some things to consider.
How hot does the dye you're planning on using have to be to be effective?
Is the suit 100% cotton.
Is the suit brand new/what is the care instructions on it?
What fiber is the lining?
What color is it now and what color are you dying it?
Most dyes require more heat for darker colors. You can juggle the failure of the fusible interfacing's adhesive against the heat required to get a dark color, by keeping the dye bath just under what would cause that failure.
For the fusible interfacing on cotton, I would guess they used the highest iron-on temperature recommended for cotton. On a household iron the cotton setting is 193 degrees F, commercially set fusible interfacing probably uses a similar temp as not to scorch the fabric.
So if you get you a candy or deep fry thermometer with measure at least down to 100 degrees F, and keep the dye bath warm enough for the dye to be effective but below 190 degrees F or so that would hopefully allow the interfacing to remain attached---might experiment on a similar garment in plain water as there's really not an "inconspicuous spot" of an interfaced area on a garment.
Interfacing is applied to the facing of a garment, not to the garment itself, and the primary reason for using fusible interfacing is to avoid the expense of the labor it would cost to sew the interfacing on. Once the garment is complete, I can't see that the failure of the adhesive would affect the performance of the the interfacing too much.
There will be shrinkage of the cotton. The higher the percentage of cotton, the more shrinkage there will be. It will shrink some just in a regular washing, more in a hot dye bath and if dried in a dryer. Fabric shrinks more on the lengthwise grain, which runs, unless there were some special circumstances or for design considerations, top to bottom of garments. So jackets, skirts, pants, the shrinkage, if noticeable at all, is going to be more noticeable in their length. So if it did shrink and the lining started hanging out the bottom and the sleeves, wouldn't be too much trouble to adjust it as with most lined garments, hanging-out-lining happens naturally with wear anyway and has to be adjusted.
It is difficult when you're working with a one-of item and can't experiment too much, but if you can find a similar garment in the back of your closet or at a thrift shop to experiment on, I'd recommend it.
Of course, all of the above is moot if you're dying a white suit to a pale baby blue
Shylaah