Measure your box spring width and length. As long as it will fit inside the platform bed edges, there is no reason you can't use it. You should also measure the height of your combined box springs and mattress. The platform bed with headboard you show is very low. Add the height of your mattress set to the inside lip height to figure how high the top of your bed would be. My mattress set is 20" thick. Were I to put my box spring/mattress on your choice of frame only 17" of the headboard would show. (6" to lip plus 20" mattress set subtracted from 43" headboard.)

How much headboard do you think you need to show? Is 17" not enough? Is 24" or 28" okay?

How would you keep a box spring plus mattress on top of a platform bed from looking weird? Just a fitted sheep on top of the box spring?