> my lab, which is just a bunch of people with laptops.
OK, so it's something akin to Wikipedia's description of "dry lab".
> The situation I'm dreaming of, is, for example, it not being my problem when a grad student is having trouble getting software installed on the server. That would be something I'd love a lab manager or dedicated programmer to deal with
This sounds very much like IT/Helpdesk support, of the kind a programmer or a lab manager typically wouldn't want (and might not be qualified) to deal with, either.
Is there something unique to your lab environment that you couldn't use some kind of shared/general IT helpdesk service, perhaps focused on scientific software users?
If not, there may well be an unmet need that a startup could fill, especially if everyone in each lab needs identical support and they can combine it with some kind of cloud-based service with basics pre-installed. However, if you're looking for unlimited, individualized user support for a limited, flat fee, that's not sustainable.
> I don't have nearly that level of funding
For a service provider, the question then becomes, do you have enough funding for the amount of service you'd require (plus the provider's overhead/profit), even if it's not nearly enough for a full-time person (which can be remarkably expensive in popular tech hubs)?
Reasonable IT support to user ratios are somewhere in the 1:20-1:100 range, which is the same range (for a 3-person lab) as HappiLab's pricing.