Is there a way to gauge how clogged the EOS resource exchange is where transactions would get stuck?

You can use the following command line argument:

cleos -u $eos get table eosio eosio rexpool

This will show a result something like this:

{
  "rows": [{
      "version": 0,
      "total_lent": "47137407.6385 EOS",
      "total_unlent": "20952015.1375 EOS",
      "total_rent": "11766.0122 EOS",
      "total_lendable": "68089422.7760 EOS",
      "total_rex": "677469852402.5966 REX",
      "namebid_proceeds": "0.0000 EOS",
      "loan_num": 256463
    }
  ],
  "more": false,
  "next_key": ""
}

Here you can see how much EOS is being lent, and how much is still lendable, etc.

There are also various websites which show REX statistics, such as https://bloks.io/rex?tab=pool