Because the machinery of the bureaucracy loves more than anything the status quo and despises change or innovation, the burden of proof will always be upon your shoulders. Unless you seriously muff it, you will be able to negotiate at least limited accessibility to suitable turf. After all, you are petitioning a public entity, and you represent a responsible segment of the public, proposing an appropriately broader use of the publicly owned lawns. At this early stage of negotiation, you will want to promise as little as possible to get as much as you can. This is easy to do.
Precedents vary depending on the locale.
Cities with budget problems may respond only to cash donations or pledges of future financial support for maintenance. Other municipalities may designate specialty lawn facilities for croquet without requesting any payment or pledge, so long as some form of public access is guaranteed. A few are even required by law to allocate public recreation facilities in proportion to public demand. Since you must raise money to buy and maintain equipment, if nothing else, you may as well settle on membership dues high enough to discourage people with no more than a casual interest and low enough to attract the kind and number of membership you want for your club and for the available turf. The principal avenue of access to new membership could be through publicly advertised introductory courses.