As you indicated, spherified caviar will solidify after 20–30 minutes as the sodium alginate gel reaction continues until the entire sphere hardens. This will take correspondingly longer for huge spherifications. When it comes to leaving, there are two things to consider: bacterial contamination and syneresis.
Spherified caviar should be treated like any other perishable food for health and safety concerns. They could be kept at room temperature for a maximum of four hours. They will keep longer if kept chilled. Alginate gels may be thermo irreversible depending on the concentration of calcium ions, which is usually low for spherified caviar (they are not at high calcium concentrations), so spherified caviar could theoretically be kept in a suitable bath at a higher temperature, say 70C, which would extend their shelf life. In general, handle caviar the same way you would the base it’s manufactured from in terms of food safety.
Syneresis is a process in which alginate gels leak water over time. This can change the structure of the gel and have an influence on other components of a dish that includes alginate gel. Once created, the easiest approach to avoid this is to keep your caviar in a bowl of fresh water or the same liquid (e.g. fruit juice) that was used to make the base liquid. When needed, the caviar may be taken with a slotted spoon and utilized as needed. Note that the liquid used to preserve the spheres should not contain any alginate, since the calcium from the spheres would cause it to gel, making the spheres difficult to handle.
Instead of using alginate, have you explored using alternative agents like Agar Agar droplets in cold oil to generate solid caviars?