Depends what the "pushing" means :) How much you're going to expand? If not much, the lightest possible (but old) browser is AMosaic. If expanding to little better CPU with some memory expansion, then any of the most common Amiga browsers.. I prefer IBrowse 2.4, but AWeb and Voyager are options too. They lack CSS, so they work better with early 2000s sites (you might want to try wayback machine or so) or few nice sites like this which have remained compatible :)
If you really push the upgrades, then maybe even Netsurf or OWB, which are really heavy for Amiga, but support the more modern standards like CSS.