at most professional institutions much or all of the telescope time is competitively bid. Researchers submit proposals to the Time Allocation Committee (TAC), and if they find the proposal worthy of telescope time and can fit the research into the other scheduled research, it gets approved. Some institutions are state funded and may provide for any researcher to apply for telescope time. This means professional, university or grad student, and amateurs as well. Some are only open to people from that state, while others are open to anyone with a good proposal.
Also be aware that many telescopes that are available for amateur research are not the most highly automated telescopes. Often the instruments available are older and have not been upgraded in decades. You won't be able to operate them remotely, nor do they have a dedicated telescope operator available. That means you will need to go to the telescope and get checked out on it by someone from the institution. Once that is completed you will continue your run by yourself, or with an assistant that accompanies you. Future runs after that are easier to get time approved by the TAC and there is no need for additional training again, so you can use all of your time gathering data.
My advice for you is once you have a research project in mind, you should then look at the various observatory sites that have telescopes and instruments appropriate to your program. If you can work with their standard instrument, or can schedule your time on the telescope for when they have the instrument you want on the telescope already planned, eliminating an instrument change just for you, your chances of being approved goes up.
Contact the various TAC Chairs and inquire as to the availability of the telescope for amateur research, what it would take to get qualified on that telescope, on-site accomodations, etc., etc. Note too that a project that can be productively accomplished during bright time has a better chance of being approved simply because there is less demand on the telescopes during that time of the lunation. Even if you need dark time for most of your project, requesting bright time for your first run may be helpful in getting you trained and established with a positive reputation at that observatory.
Many observatory site/telescope schedules are available on public facing web pages, so you can also see which telescopes are under-subscribed. That may be because there is an issue with the telescope or instrument, and the local researchers know to avoid it. However, could be that it is a fine telescope & instrument but it just doesn't fit anyone's research at the moment. I'd suggest that either of these are perfect opportunities for amateurs who are interested in using larger telescopes for their research.
That should give you enough to get going if you still would like to use professional instruments. I think once you start looking you may be surprised to see how many sub-meter to 2-m class telescopes are out there, and potentially available to interested and motivated researchers.
Good luck in your search and research. May I ask what you are hoping to do that requires a larger telescope?