Taxi after midnight in Nagasaki

Could you please advise on the ease of getting a taxi after midnight in Nagasaki, specifically from the Huis Ten Bosch theme park following the fireworks display? Additionally, is it possible to pre-book a taxi for this purpose? It appears that all hotels in the vicinity are fully booked. Thank you for your assistance.

When are you going to Huis Ten Bosch?
There are plenty of days for fireworks.
https://english.huistenbosch.co.jp/event/fireworks/

By the way, when I checked the fare to Nagasaki Station on a taxi site, it was about 29,560 yen (about 70 minutes).
https://www.taxisite.com/far/anytoanyret.aspx?departure=33.0861145%2C1...
There are five taxi companies.
https://www.taxisite.com/cal/pref/42/202.aspx
This company also has an English website, so it may be easy to make a reservation.
https://lucky-group.co.jp/en/
Probably, you will have to pay the fare until the taxi comes to Huis Ten Bosch in addition to the taxi fare you paid earlier.

Huis ten bosch to Nagasaki will be very expensive; it will be way cheaper to stay at a hotel around the park and come back to Nagasaki via JR next morning.

Sorry, missed the "all hotels are booked part"; I'd suggest trying to look for a hotel in Sasebo, which is way closer than Nagasaki city, a Taxi to Sasebo will be around 6000 yen. If failing everything else, the information desk at the park or the reception of the Okura hotel can probably arrange a taxi for you, or you may even consider renting a car for your visit.

For late and early hours,
taxi fares are set higher (roughly 20 percent more).

I suggest you check hotel room vacancy in a broader geographic range.

I stayed in Sasebo earlier this year - even the "best" rooms were cheap. Between a late night taxi and the room, you'll probably be less than Nagasaki.

Sasebo is a good call. Add in a visit to the Yumihari Park Observation desk in the morning as a bonus.

"observation deck" Sheesh. I should learn to type

Hello
Thank you all for answering my question, much appreciated.

I should mentioned that I booked a hotel in Sasebo, I googled the distance between out hotel and the theme park is about 15-20 minutes away by taxi.
My concern is the availability of taxis from Huis Ten Bosch after midnight to take us back to our hotel in Sasebo.