You could do this using a few built in things in Lighthouse Studio.
The first would be to build a quota (call it MyQuota) with three cells and each cell having a limit of 60 respondents. Rather than logic, you would have each cell set to always qualify and then set the quota to check for cell membership randomly. This will basically randomly put someone into the cell and stop putting people into a cell once it fills up.
Once we have a 1-3 assignment coming from the quota, we can use the randomized blocks capability in the software. First, build a list (call it ParentList) that just has 3 items in it. Next, build a constructed list (call it BlockList) with the instructions like this:
What will happen is MyQuota will evaluate to a 1-3 value, which will add the corresponding item from the parent list to the constructed list.
Next, set up randomized blocks and then tell the blocks to be controlled by the constructed list. This way the respondent will only see the one block that is controlled by whichever quota cell they get put in to.