I was wondering if there is some penetrating finish that will harden up the outside a little bit.
"Danish oil" and other mixed oil-and-varnish blends will do this, but you'll only get a modest improvement. One source claims it makes the surface of wood 25 or 30% harder but that has to be either a figure directly from a manufacturer or an estimate of the author. In either case expect this to vary because "Danish oil" varies significantly from brand to brand (some have more varnish in the mix and those will be harder).
Varnish will toughen the surface too in a different way because it sits on top and doesn't penetrate as much*. Thinning the varnish heavily will aid penetration only a small amount (much less than one would think) except on end grain, but it is reasonably common practice to dilute the first one or two coats and then go to straight varnish or varnish thinned only a small amount for the finishing coats so if this might help penetration even slightly there's little reason not to try. If wiping the varnish on instead of applying by brush or roller the above will occur naturally anyway.
general finishes water based poly (high performance).
That's not a penetrating finish. One of the reasons for the classic lighter colour and less enhancement of figure imparted by waterbased polys is they penetrate the surface of the wood less than their oil-based counterparts (they're said to 'wet' the wood less effectively).
Even though finishes of this type are not at all like oil-based varnish they will still harden the surface in a similar way, although usually not as much.
I also wanted to paint the chest. Would I paint before or after the water based poly, if the main purpose of the poly is to harden the wood a little bit?
Generally if you're going to paint and varnish the varnish would go on after the paint. This isn't a hard-and-fast rule though and there's no technical reason you couldn't paint over wood that has had waterbased poly applied (particularly if it's just one thin coat) and then apply one or more subsequent coats to add a protective coating to the paint.
A full, glossy varnish film would present an adhesion problem for paints in exactly the same way that they'd have difficulty sticking to glossy plastic (because that's exactly what hardened waterbased poly is, a type of plastic). But a thin coating on wood presents a surface with abundant texture, from microscopic to macroscopic, which paint should have no problems adhering to. If in doubt though, very lightly scuff or sand before painting.
*Wood in general doesn't want to allow things to penetrate, it's not really in its nature. This is why after staining it's so easy to accidentally sand through to bare wood, and that's with stains specifically made to "penetrate deeply" as the manufacturers all love to tell us! So, on face grain, the difference in penetration between a penetrating finish and oil-based varnish would actually be very small in fact, a difference of only a couple of thousandths (hundredths of a mm) at best.