Superconducting quantum technologies require qubit systems whose properties meet several often conflicting requirements, such as long coherence times and high anharmonicity. Here, we provide an engineering framework based on a generalized superconducting qubit model in the flux regime, which abstracts multiple circuit design parameters and thereby supports design optimization across multiple qubit properties. We experimentally investigate a special parameter regime which has both high anharmonicity ($\sim\!1$GHz) and long quantum coherence times ($T_1\!=\!40\!-\!80\,\mathrm{\mu s}$ and $T_\mathrm{2Echo}\!=\!2T_1$).