Work down the constraints in this order, and the answer usually falls out before you reach the bottom.
1. Is there a hard rule about where the data may go? A regulation, a customer contract, a classification. If yes, that eliminates options before anything else is considered, and no capability argument overturns it.
2. Does the system have to work without connectivity? A factory floor, a vessel, a remote site, an isolated control network. If yes, you are on-premise or air-gapped and the rest is detail.
3. Is the volume sustained and high? If yes, fixed-cost hosting will beat per-token pricing, and the crossover is worth calculating properly rather than estimating. If the volume is low or spiky, the API wins and will keep winning.
4. Does the workload need frontier capability for every request? Rarely true once tested, and if it is genuinely true then hosted is the answer. If it is true for a minority, that is a routing problem rather than an infrastructure one.
5. Is there anyone to operate it? The question that quietly decides more of these than the other four combined. If there is no team and no budget for managed operation, choose the option with the lowest operational burden that satisfies your constraints — and be honest about it now rather than in eighteen months.