Theoretically... if you have a (trusted) "friend" that has a full unpruned node, one can simply copy the blocks/chainstate and drop them into a new node installation and "voilà", you have your full node without having to download anything, you just have to trust your "friend".
You can even set the new node to be pruned (after you have loaded and scanned your wallets) and it will discard all the unneeded blocks.
Of course, if storage space is the issue (and not bandwidth)... then this might not be an option either.
I just want to run a full node where I can just type in a certain public key, and it tells me which UTXOs are associated with this key. Basically a local version of blockchain.com/explorer
In any case, the OPs requirement to "be able to enter a random public key and view the balance" might preclude pruning anyway... as most (all?) blockexplorer apps require
txindex, which is not compatible with pruning.
Alternatively, a bespoke software solution might be possible.