The "right" thing to do is to invest that money over the long term in something that will most likely yield far more. A typical stock index fund, for example.
It sounds like that is off the table, and 3.8% > ~1% from a savings account. Certainly not the worst use of the money, but sub-optimal.
We actually did do almost exactly this back in 2014. Wound up with about $50K larger mortgage than we originally intended due to a delay in closing on the "old" house. When the transaction cleared a month later, made a $50K extra principal payment. Now, we were on the Dave-Ramsey plan and only just starting to learn about MMM then, but looking back, had we invested that, we'd have about $10K more today. And that was only 3 years ago. We also paid off the remaining mortgage early, finishing this past June.
So, knowing what I know now, I would have pushed to do neither thing - not the extra $50K, and no extra payments either. Also would have looked at a 30 year instead of a 15 year. But, luckily, we're in a LCOL area and the house is a small part of our financial world, and getting smaller every day. The lower cash-flow is nice, but having the house paid off is actually costing us money.