No! Vinyl plank will fail if installed directly over hardwood flooring. You need an approved underlayment panel. I have had good success with Multi-ply, accu-ply Halex, and FiberRock.

You had a THICK engineered hardwood before installed directly over the plywood substrate. Now you have a THIN 6.5mm vinyl floor (cork attached). But you have had to add cement board over that subfloor because you needed a floor height raise to get the doors and trim to work. Your old engineered was NOT sitting on cement board (which is not structural and would need WORK....patching...sanding...etc). So the comparison between one floor and the other isn't fair...there have been plenty of changes. Subfloor preparation is separate from installation labor. It can and SHOULD cost $1 - $5/sf for subfloor preparation (not including the cement board). How much was installed? Are there transition strips between doorways? How much space was left at the walls? A bit more information would help. And normally you do not glue down a floating vinyl floor. This particular one has NO INSTRUCTIONS for adhesive. That's a very TELLING sign. If the planks have cork backing, then normally are NOT glued. Your best bet is to either live with it or pull all of it and have all the prep done and then relay it. But be aware that you will lose as much as 25% - 50% of your planks because the click edge will break.

I would ask why such a steep price for installation. Ask which order the kitchen will be put back together. A small room installation cost could work out because they use a "day" wage for small projects. I've heard of minimum charges of 1/2 day of work or 1 full day of work regardless of how long it takes. But I'm suspicious of this because the installer has MANY other things going on (or so I believe by your post). A person who is redoing someone's kitchen will often give a little so they can gain the entire project. The small space install only works if the ONLY THING s/he's doing is installing. And I never buy the 'nothing else needs doing' jazz. A sheet vinyl floor can have some WONKY substrates underneath (can look like a roller coaster ride). Sheet materials are VERY forgiving. A plank vinyl is NOT forgiving...at all. In fact, a poorly prepared substrate is GUARANTEED to void your warranty. And I mean 99.3% of the time the lack of prep is going to void your warranty IMMEDIATELY (before he leaves for home). I think you have some conversations coming up in the next few days. And to be clear for us, is this the SAME person who is doing the ENTIRE renovation?

At this point, the job ($4/sf for a BIG job with LOTS of things to do = awesome price) is finished. The cure is going to be worse than the disease. I'm a big fan of "Wait until you have a problem before throwing money/time at it." It is the cheapest solution I know of (I'm a pragmatist...who is both cheap AND lazy ;-P). My advice is this: leave it alone. Keep 10% 'extra' of the flooring product (roughly 75sf or to the nearest box). This will offer you the peace of mind and the material security you will need to feel secure. You will keep it in the back of your mind. You will know that IF something goes wrong you have ALL THE PRODUCT needed to repair/replace the damaged planks. With a laminate floor I wouldn't worry to much about a bit of bounce. People have 'put up with' bouncing laminate floors for 20 years (the SAME laminate they installed 20 years prior). That tells me the click system is fine. Your floor is a rigid vinyl (8mm thick = nice product). The rigidity of the core will help keep everything stable. The only weak point (as in everything in life) is the link (click edge) between one plank and another. This is the one and only place things *might go wrong. Now to be blunt, a rigid vinyl floor like yours can be taken apart and, "in theory", be clicked back together - BUT I don't like that idea. I really don't. We know the click-edge is delicate. We know that. We've seen them brake by grazing the ground before being installed. A vinyl floor that has been disassembled has a STRONG likelihood of DAMAGED edges. If the edge is damaged then there is almost no way you will get them back together. That means you will have to assume 25% damage should you choose to 'unzip' this floor to the point where it bounces. That leaves you with 75% intact flooring with 25% garbage planks (which must be replaced). As soon as you try to fix the subfloor underneath (by removing the flooring) you will have lost the 'intact' floor. You are guaranteed to spend MORE time, MORE money and MORE materials to deal with this. Compare that to 'leave it alone'. Right now you have 100% intact flooring. Every day you wake up to 100% intact floor is another day with a perfectly functional floor. Every day you have 100% intact floor is another day you have SAVED time, money and materials. Simply leave this floor alone until something MAKES you repair it. Keep your 5%-10% extra on hand for future repairs (and there will be future repairs). Use them if and when you need them.