Hiring the experienced pro isn't the only option. ...If you can find someone like that it gives you a nice position in-between the cost of a full-time pro vs the quality you're likely to get if you don't have someone on set specifically experienced with audio work.
IMHO, this is excellent advice for the more advanced/serious amateur filmmaker. Essentially, do the best you can with the budget you've got. In other words, make a film/short as close to professional standards as one can get by employing the best, most experienced amateurs/aspiring pros one can find to cover those areas where DIY is not practical. Progression from newbie amateur to skilled amateur is therefore mainly concerned with: Developing one's own DIY skills in that/those filmmaking areas of greatest interest, identifying those other filmmaking areas of greatest concern and finding people to fulfil those roles, develop scriptwriting and/or script adjustment abilities, develop planning techniques and workflows (often "on the hoof") which enables the very difficult task of actually completing a project AND, achieving all of this with the minimum number of unacceptable compromises, within a budget fixed before starting, typically 100-1000 times smaller than the budget of an equivalent professional/commercial project.
However(!), while some of these fundamentals of more accomplished amateur filmmaking maybe related or have some similarities with professional filmmaking others are not. In other words, improving one's amateur filmmaking abilities means years of study, practise and "trail and error" learning, a large proportion of which is of little or no use in professional filmmaking. In other words, if one aspires to professional filmmaking, some of those years spent becoming a skilled amateur have been wasted in the gaining of unnecessary knowledge/skills, while many of the essential knowledge/skills specific to professional filmmaking have been ignored, to the point of not even being aware of their existence!
Baring in mind the OP's username (and stated aims) and for the reasons I've just discussed, I don't believe that going for the "in-between position" or the basic philosophy of trying to make the best film one can (with effectively insufficient resources) is the best long term strategy.
Assuming we can get a cheap locale, the total cost would be $250-$300 max ... Would that be a reasonable budget?
A reasonable budget for what? What is it you are trying to achieve?! ... If you say "experience" then I'll just ask; "experience of what"? If you want to gain knowledge/experience which takes you towards professional goals, then you've got to start thinking professionally and develop a professional approach to achieving your goals! ... $250-$300 could be entirely "reasonable", so could $100, so could $20,000. ... It's like asking if $250-$300 is a reasonable budget for a computer; it's an entirely reasonable budget for a little laptop to browse the web and reply to a few emails but if you want to edit a 4K theatrical feature you're not going to get anything useful for $250-$300, you'd need to budget more than that just for hard drives! In fact, it's nonsensical to ask that question in the first place without providing a decent indication of what you wanted the computer for!
So again, what is it you want to achieve? For example, gain some basic experience of organising and managing a team of people? Or is it to experience professional workflows, the interaction between professional film crafts-people and managing them? Your budget would be entirely reasonable for the former but entirely unreasonable for the latter!
I have to say, the role of Producer is IMHO, the most difficult of all the filmmaking crafts to learn via the route of starting off as an amateur. This is because it's often arguably the least important individual role in amateur filmmaking and arguably the most important individual role in professional filmmaking! As a professional Producer you need to ask (and obviously answer) questions such as: What is this film for, for what market and for what platform? What resources are required to guarantee exceeding the minimum market expectations? What is the ratio between the cost of those required resources and the realistic gross returns? How can you affect that ratio enough and provide enough risk management to interest investors in providing the funds for those required resources? And of then of course, managing the project to ensure an end product which meets it's quality targets and achieves the projected investor returns. In many respects this is the complete reverse of amateur filmmaking, where the budget is set and then everything (resources, final quality, etc.) is essentially a lottery which revolves around that budget. IMHO, you need to apply this same basic professional approach even to your current situation! What, precisely, is the minimum you wish to achieve from making this scene/short? What resources are required to guarantee (at least) meeting those aims/targets. How much will those resources cost? Is the ratio of cost to benefit (meeting your target/s) worthwhile to you personally (as you are also the sole investor). How can you affect that ratio without significantly increasing the risk to the guarantee of meeting your target/s?
G