Staying completely out of politics and sides - I worked for the Army for 30 years and EVERY year that we didn't have a signed budget
(which was a lot) we were put under first the threat of a Continuing Resolution Authority (CRA) which meant we would be funded for a short time under last years or the last year a Budget was approved, second a shut down. Almost every year we had to plan for both which required a lot of meetings, briefings, due outs, etc...
Long story short - Each Federal Agency has the authority to operate under a budget or CRA as they wish and under a shutdown each Agency still makes its individual decisions on what is and is not critical and or essential. IF BLM determines it is in the public interest to maintain operations they can, (even without a budget and even during a shut down). If they (BLM) decide it makes no difference to the public interest then they can shut down.
There are of course sacred cows (which I was one of - Police, Fire, EMS, DR's, Nurses, Physical Security, Intelligence, etc...) Also a sacred cow is contractors, the reason for this is the contracts are already signed and in place and have already been funded so not approving current or future funding doesn't (CAN'T) effect current contracts. If it did the Federal GOV would be in violation of the contract and subject to litigation for not meeting the contract and negatively effecting the contractors, its business, its reputation, its ability to meet their funding obligations, etc... It truly becomes complicated fast but all in positions of responsibility for budget oversight or obligation know what is and is not required.
I will close by again stating - the decision in what is critical and or essential during a GOV shutdown resets with each individual Agency. So with that said I doubt very seriously if the guy at the check in booth knows anything other than if they will come to work on Monday if there is a shutdown and nothing else.