Why not give gpt-5-mini (via the response API) a shot?
You can use the “instructions” parameter as your system prompt - I use it to give output formatting instructions (among other things). It works quite well.
Of course, you can change the highlighted parms.
{“model”: “gpt-5-mini”,
“tools”: [{“type”: “web_search”, “search_context_size”: “high”, “user_location”: {“type”: “approximate”, “country”: “US”}}],
“reasoning”: {“effort”: “medium”},
“text”: {“verbosity”: “medium”},
“instructions”: “Put your system prompt here”,
“input”: “Put your question here”
}