Extira is a nice product but just like Joe said it will not hold fine detail.
Here is my $0.02
Looking on your design you are going to pocket a lot of area. That exposed core is lower density, more porous and much more sealing has to be done.
You may be able to hold detail if you coat surface before machining to bond fibers with sealant that will penetrate top surface but I have my doubts.
How about MDO faced with acrylic or pine glue up and painted.
Or
Do it as they want just have to explain material choice is not ideal but you work with given budget etc. have them sign proposal with verbage that this will carry so and so warranty due to choice of material by client etc.... so you are covered.
Produce sign they will be happy with quality and schedule, you collect money.
How many times I heard that they do not have money in budget.......
They want Cadillac for price of Yugo I am not charity and go upside down on all my projects. I work with client and negotiating is part of the job.
Look at my signature
100% MADE IN TEXAS! It's not part of China partner.....