I'm working on a video game project with a friend that would thematically take place during 4AD with a focus on Gladiatorial games. I have a goal of making the game's atmosphere as close to real life as possible and to recreate the feeling of how it was to participate in one of those games from the perspective of the gladiators. I have myself did some research. mainly with resources by Garrett G. Fagan, but still I feel like a life consultation with an expert would greatly benefit to the accuracy of the project.
As such I'm in need of a history consultant knowledgeable about gladiators and gladiatorial games, but I'm at an in-pass as I have no idea of how to go about the process of actually finding an historian/expert like that, especially since we are a tiny indie game studio and with a very limited budget.
So if you're asking "is there some agency or something that handles all of these kinds of requests?" the answer is, "no, not really." The demand for historical consultants is just not that high (sadly). I am aware that over the years there have been a few attempts to organize such people as formal consultants but I would not expect such services to be very comprehensive; most historians don't do anything like this on a regular basis.
The way it typically works is, if you are looking for a consultant, you do a little research into the relevant books/articles/public appearances of people working in that subfield. If you find someone who seems to have the right knowledge, you look them up on their academic website, and you send them an e-mail. Maybe they'll want to work with you, maybe they won't — but if they don't, they'll probably send you somewhere else.
When I've done historical consulting (for both fiction and non-fiction projects), it's worked like the above, and what I always do is first have a phone call with the producer about what they have in mind. What I am interested in knowing is a) what's the project (is this something I want to be associated with?), b) what kind of expertise do they actually want (it may not be what I do — e.g., there is a big difference between "historical events" and "historical window dressing," like costumes, and I don't do the latter at all), c) what time commitments are they asking me to give (is this a "I could resolve this in one phone call or e-mail" level of consulting, or is this "I am essentially working for you part time for a year" level of consulting), and d) what kind of compensation are we talking about (for not-for-profit causes doing work that I think is socially valuable, I'll do some consulting for free, if I have the time; for for-profit endeavors, especially ones that are asking for a lot of my time, I will expect to be compensated). Just as an indication of what a historian might be thinking about.
You speak of your "very limited budget," which I'm sure is the case, but I would just give a gentle reminder that if you are looking for academic experts on a subject, these are people who have devoted decades of their life to subject-level mastery. It's OK to lead with the "we don't have much budget" line, but you should be aware that if you want an expert, they will expect to be treated and paid like one, if you are going to take up any amount of their time beyond a conversation or an e-mail (most historians I know would allow you to take up that amount of time for free, but much more beyond that cuts into the other activities we are obligated to do).
Good luck!