Sorry, I don't have direct experience booking private car from HKG to TST, but I do take a private car service from HKG to/from Shenzhen. This is booked by my company so I don't have the direct contact information but it seems that the service is probably similar service to what can be booked with Klook. I would venture to guess that the same drivers I use offer themselves up for Klook also. My pickup/dropoff is handed with a Toyota Alphard van (sometimes a Vellfire). For 4 people with luggage, I would assume this would be the proper choice for you. The driver is typically waiting for me outside Arrival Hall A with a sign and takes the bags for me and we walk out to the car park area.

My company pays 600 RMB (~680 HKD) for pickup at HKG and dropoff in Futian area of Shenzhen through one of the checkpoints, Shenzhen bay or Huangang. If I arrive early morning, before 7AM, the price gets bumped up to 700 RMB (~800HKD). A trip during one of the big holidays, LNY, National Day, Labor day, the price might be 800RMB since many drivers are off for the holidays. I would think a trip from HKG to TST would be considerably cheaper than the 700 HKD I pay for transport to Shenzhen.

I am not familiar with licensing requirements and whether these drivers are licensed but it is all prearranged by my company. I do pay the driver personally and charge back to the company due to the way my contract works.

I've never used the Klook advertised service, but again, I wouldn't be surprised if the same drivers I use also work with Klook. 1350HKD seems pretty steep but it is a hotel prearranged service and hotels are notorious for overcharging for these sorts of things.

The cars these guys use, Alphards or Vellfires, are all clean and in new or nearly new condition. The drivers are professional although I've had some that can drive a bit crazy at times. I would have no second thoughts about booking a pickup through Klook.