Battery voltage dropping upon connection

You are going about this backwards. You need to first figure out what the current demand is of the device you are trying to power. Then you need to determine how long you want it to run. THEN you can look at what battery you need to give that performance.

You are trying to run a diesel locomotive on a scooter engine.

You are going to have to either use a bigger battery, or eliminate functions until the battery you want to use can support the load.