The shift you are aasking about is specifically Hatha to Iyengar?
Just for a second let’s set aside classical definitions and so forth because we’ll easily get mucked up in that.
Iyengar yoga (depending on the teacher and the work they’ve done) is a method. It is alignment-based and to some can seem meticulous. Hatha yoga (in the U.S.) works as more of a catch-all term for physically based yoga. Thios is NOT the classical definition of HaTha. It is however it’s common application. So it is likely you will get a bit of variety from one Hatha instructor to another. In an Iyengar practice you are more likely to get commonalities from one instructor to the next. There are some things done the same way each time and other things not dane at all each time.
With Hatha, as it’s applied here, you can get one teacher instructing Sarvangasana on a bare floor and another requiring props.
I don’t think the transition would be that tough depending on where you are in your practice.