Problem flattening thin boards

Double sided sticky tape.

Well, You can just sorta live with it and maybe you can force it into place with your joinery . Or you can get thicker pieces with as much dimensionally stable traits as possible and work your way down to final thickness slowly. And my slowly I mean , take some off and sticker it and come back in a day or 2 , but this is extreme unless it's required. Or support the crown or cup from underneath with a handful of shavings. I personally wouldn't bother . Hopefully whatever you're making lends itself to working the cup out by force . As my machinist friend would tell me " I can't stand wood , it keeps moving! "

I would try shimming the concave side while you flatten the other face, then flip and plane again. But you’ll lose some thickness.

Before you spend a great deal of time flattening your wood, you should understand that plain-sawn boards tend to cup depending on the orientation of their annual rings and the relative humidity of their environment. This has to do with the difference in which plain-sawn and quartersawn grain expands and contracts. Plain-sawn boards, which are cut tangent to their annual rings, expand and contract up to 8% of their width across the grain with changes in relative humidity. Quartersawn boards, which are sawn radially to the annual rings, expand and contract just 4%. (These are averages; different wood species have different rates at which they expand and contract. But 8% and 4% is a good rule of thumb for planning purposes.)

Because the orientation of the annual rings changes across the width of the board in plain-sawn lumber, different parts of the board will expand and contract at different rates. The rings in a typical plain-sawn board are tangent to the surface at the center and move towards radial at the edges. Because this, the board expands and contracts more at the center than at the edges. Furthermore, there is a difference between the surfaces -- there tends to be more tangentially-oriented grain on the outside or "bark" side of the board than on the inside or "pith" side. Because of this, the boards will cup in the opposite direction of the annual rings as they shrink, and in the same direction of the rings as they expand.

Relative humidity changes not just with the weather, but also with the environment. Your shop is probably drier than the lumberyard. Consequently, the lumber will be shrinking when you bring it into your shop and the plain-sawn boards will be cupping. The same is true when you finish a project and you move it into your home -- the wood will be in motion once again as it adjusts to the relative humidity of its new environment. Your boards may cup further or flatten themselves.

All of which is to say that Dmacnutt had a very good point when he said you can keep the wood flat with joinery. Intelligent wood joinery anticipates the motion of the wood, allowing it to expand and contract and keeping the parts relatively flat and straight. If the cup will interfere with the building process, I will true a surface on my jointer, then plane the other surface parallel to it on my planer. But I plan the joinery to keep it flat after I drag the completed project out of my carriage barn and into the real world.

I read spraying the cupped face with water and letting dry can help, since cupping is caused by one face drying faster than the other

I keep a deck of playing cards laying around for shims.