One reasonable reschedule request is usually not a major issue, especially when you ask early and the conflict is legitimate. Recruiters deal with scheduling changes all the time.

The way you communicate matters more than the fact that you asked. Be concise, appreciative, and flexible. Give enough context to show that the conflict is real, but you do not need to provide a long personal explanation.

It also helps to offer a few alternative windows rather than simply saying the proposed date does not work. That makes the request easier to solve and signals that you still care about the opportunity.

Do not let fear of looking difficult push you into an interview date that genuinely creates a problem. A professional reschedule is usually better than showing up distracted, late, or under unnecessary pressure. A useful reference is writing a professional email to a consulting recruiter if you want to go deeper on that part.

I rescheduled once and nobody cared at all.

Just give them as much notice as you can. That is the part I would worry about.