I have two methods to propose, and I shall post them as separate answers (so that people can upvote/downvote them separately).
First, I remark that trying to estimate player strength under 10 minutes is not a good idea unless you intend to train them specifically for blitz (not everyone work well under such time pressure). Hence, spending 15 or 20 minutes for this is required to avoid feeling too much pressure (and this time is worth spending since measuring strength of the audience before doing other sessions is important).
The first one is similar to tactics puzzles idea *[Note1].
Instead of giving tactics puzzles that have a right solution which is probably tricky, give positions and ask them to give their moves if that occur in their games. In those positions, there may or may not be a tactical resource. The side to play may even be lossing. The idea is to give positions that simulate actual game situation, and to test judgement in addition to caluclation ability.
My suggestion is the "What's your move" exercise (appears as Practice (3) in this answer; the usual tactics training method is Practice (2) ).
A set of 6 or 8 positions are given. Each student should select 4 of them and answer the question "What is your move in this position?" for those four. A position in the set may or may not involve a tactic, and it may have more than one answers (e.g. depending the plan the player comes up with). This simulates the game situation more closely. I suggest to include easily winning positions, clearly lossing positions, calm positions, and so on.
The player has to decide which positions require a more thorough calculation. There is no guarantee that there is tactical resource for you (e.g. no automatic sacrifices). Deciding when to calculate deeply is a crucial skill for a serious chess player.
I also suggest to include positions with different point values for answers. E.g.: there may be a hanging piece, a royal fork and a quick checkmate available. The move that initiates the checkmate is the best move and gets full points (say 5 points), still the royal fork gives 4 points, and capturing the hanging piece gives 3 points. The bottom-line is not to evaluate answers as binary (best move or not).
A set of positions similar to this idea is available at this link (although the webpage cannot properly do the task claimed, this should be sufficient for your task). Of course, you need to bring down hardness level of the puzzle set to suit your intended audience.
*[Note1] By the way, the multiple choice idea is bad since it eliminates some important aspects of tatics such as creativity, and tests only others such as decision making skill.