Looks like your W is very unreasonable and is not putting the children's needs first. Why on earth would she object to them seeing a psychologist? Hypothetically, what would have happened if you unilaterally took them to a psychologist without her input? Don't be tentative. You know what is right.
Are you still talking to your W directly? I think she is very manipulative and you should let the Ls do the negotiating.
Most importantly, decide - as best as you can - what is best for kids, and do it. They are the victims here. No one will persecute you for doing the right thing. " Do the right thing" should be your mantra.
