Side spray and a portable dishwasher
Side sprayers or vegetable sprayers are a known weak point of most faucet designs.  The valve is not meant for long-term pressure and are not designed for that task.

The diverter valve does play a role, but it's not to cut off water flow from the sprayer.  The diverter's job is to cut off water from the main faucet and divert it to the sprayer when the pressure on the spray side drops (when you pull the trigger).

Normally, most of the water pressure is going out the faucet and the sprayer only has to stop a small fraction of the pressure.  When a dishwasher is hooked up to the end of the faucet and the water flow is stopped,  the sprayer then has to deal with the full pressure of your water system.

To remove the side spray, you will need a threaded cap to cover the connection on the faucet where the spray hose connects.  You can remove the sprayer and take it to the store to ensure you get the right size.  This should be reversible, so if the dishwasher is no longer used the sprayer can be reattached.