Make curved frame (already done), make slats with the curved ends (already done), dry-fit everything in place (no dowels or holes yet, already done), clamp some cross-slats on top of the 'real' slats for alignment reference, make a jig-block (shop-made dowel jig) and align it off your 'reference' cross-slats, drill holes in the slats, put dowel centers in the holes and transfer position from slats to frame, drill holes in frame, check fit/alignment (tip: use undersize slightly loose dowels), remove temp dowels and install real dowels and assemble with outdoor-rated glue. Done.
Cause it's either that or go with some stub tenons and blind mortises.

Unless you want to all out and do some wedged through tenons with maybe a contasting wood for the wedges?
Hmmm, maybe use some floating tenons and then just cut the mortises in the frame and the slats and epoxy it all together (epoxy for the gap-filling properties, maybe a 'thick' gel or filled epoxy)? A little bit 'loose' on the mortises and the epoxy will fill the gaps and be hidden anyway.
Quickest would probably just be the pocket holes underneath the slats and then fill the pocket holes with plugs. Not visible from the top or the front or back, just from underneath.