Good stove, IMO. It uses standard 11" WC pressure as do all the standard RV appliances.
- flexible hose from black pipe to regulator on bottle
IMO, It would be best to use a regulator that mounts on the trailer wall and has the rubber pigtail to connect to the cylinder. A regulator like/similar to
one of these. These are 2 stage and offer better regulation than the single stage that usually fir directly to the cylinder. The pigtail incorporates the excess flow preventer and a thermal melt down stopper. Note the regulators have a "this side down"... the vent faces down.
From there iron pipe can be used down and under the chassis. A 90 degree elbow would let you run the iron pipe up through the wood floor. Use a rubber grommet to isolate the pipe from rubbing. Not that iron through wood is likely to wear through no matter the amount of movement. Secure the pipe well and movement will be nil or next to it. Use a good pipe dope on the iron pipe fittings. better than tape. I would istall a tee somewhere to make any future additions easier. In copper tubed lines that is less important as copper can be easily cut and flared at any time.
In the USA any copper tubing fittings should be flare fittings to meet applicable codes. Different rules in some other places.
$1800 is what someone who doesn't really need the work might charge.
If you have little or no gas fitting experience a knowledgeable advisor would be good. An independent plumber might be willing to advise and check the work out. Just have to start asking around.