So most types of clubs like rocket clubs, astronomy clubs, etc...will fit within the 501(c)(3) and/or 501(c)(7) status. There are differences between them, but either is appropriate for this type of activity. See
IRS Publication 557
. Generally, any of the 501(c) type organizations could be organized as a 501(c)(3), but may benefit from one of the particular sub-organizations due to record keeping and filing requirements. The sub-organizations are more specialized and the requirements are more tailored to that type of activity.
Ancillary to IRS status is corporate filing status with the Secretary of State. While both involve paperwork, personally, I would not characterize it as overly burdensome paperwork....lol (but I guess that comes from perspective and experience)
Annual corporate filing is done online, and can be a simple as fill in a one page form. Initial corporate organization involves the most work. The IRS filing requirement is another story though...it is a matter of record keeping and can be a headache for the person responsible depending on how diligent the records are (eg. Spend a lot of time during the filing year keeping good records, or spend an exorbitant amount of time at the end of the year trying to collect all the necessary information). E-filing is possible, but since this is a specialty filing, it is not included in any "free" e-filing software packages.
I will add, that perhaps the primary reason that most groups do these types of filings is in order to secure liability protection and insurance. As Kenn alludes to, this is where being a NAR Section, sort of obviates the need to do this extra work, as NAR provided Section insurance coverage.
Disclaimer, while I am an attorney (retired) this is not legal advice and CMASS and you are not my clients. You and the Rhode Island club are strongly advised to seek out a RI attorney who practices within this area of non-profit organization law.