In very general terms, you can have a particular judge go in any direction here.
It is sometimes felt that the person seeking the evaluation should pay, but judges will often order that the cost be split in some fashion. Has the Ex petitioned in forma pauperis (asked to not have to put up court costs)?
The amounts you are quoting for an evaluation seem high.
When it comes to shared custody, that is presumed by the courts to be in the best interests of the child. The ability of one parent to pay any support would be considered fairly irrelevant to the best interests mandate.
However, if the current order is for supervised visitation, then there must have been good reasons for that and the person seeking the change has the burden of showing a change in circumstances to justify the change.
I have never much liked the trend towards evaluations. Sadly, in too many cases the judges like having the evaluation done because it greatly reduces their workload. Someone else, in effect, is doing the work that would otherwise come out in testimony and evidence at trial. There is also a wide range of competence in evaluation making.