Hard to tell from that picture. I would have to look at it in person but even though I mow them all the time for work at golf course, it's the last thing I want for the house. Way too demanding to look great.

Zoysia.... looks awesome for being cut every 1-2 weeks around the clubhouse. Ha.

Usually, they would go with cultivar type if going sod way. 419 seems to be very common these days but who knows what the options were 7 years ago.

I would just focus on maintaining bermuda the right way while slowly plugging in zoysia preferably Palisades if you can find lawns to steal sprigs from landscape beds to plant them in random places while trees get larger and larger. Zoysia will thrive with less sun compared to bermuda and it looks better too.