Based on my experience, the answer would be 'no'. Your doctor's office manager or testing coordinator should be able to deal with both insurance companies during the change. Your new insurance company cannot deny you treatment for a per-existing condition and that includes any additional testing needed to continue your treatment.
In our case, we were dropped from our first insurance company because our business did not meet the definition of a "group" under the rules of Obamacare. That was less than 1 month after my husband's aneurysm was discovered and it left us with 2 months to find new coverage. During those 2 months, the first insurance company continued to pay for all the testing needed for his eventual surgery. Coverage for the first insurance company ended on Dec.1. His surgery was on Dec.8. All bills for his hospital stay were covered by the second insurance company.
But here's the rub: when you switch insurance companies mid-year, you also reset what you paid into your deductible back to $0. So if you have a high deductible right now and have been paying OOP for a lot of your health care needs, none of that will be considered when you switch your insurance over to your husband's new plan - even if it is with the very same insurance company that you have now. If you expect to be spending a lot on medical bills under the new insurance plan, it might make sense to pay for a gold plan with $0 deductible for the remainder of this year.