If it were the solenoid, it would be the same regardless of which battery is active. I doubt this is a connection issue.
Deep Cycle Batteries don't always have the same Cold Cranking Amps as a starting battery. House batteries get drawn down deeply then are re-charged. This process will wear down a battery over time. Eventually the deep cycle will read as Full, but discharge very quickly, indicating that it wasn't actually up to its full capacity to begin with. Starting batteries have a short surge of high amp draw, then nothing and are immediately topped off by the alternator again.
What you are describing is a Deep Cycle house bank which is coming to end of its duty cycle. START on the Starting Battery and SIT on the house bank. You would only use the house to supplement starting in an emergency.