You don't need a O licence for use domestically in the UK for that combination unless your trailer is made of lead. For reference concrete pumps, compressors, generators etc come under mobile plant anyway but should a knob get penicity if the lighting is affixed to the trailer chassis and the plant can be lifted off (whether it's bolted down or not) then the trailer chassis is the trailer.... The plant becomes a load.
You can go whatever radius you like within the UK without O licence with a 3.5t or less towing vehicle dragging a trailer...... However tachograph laws and driving hours are a different kettle of fish.
With regards to one of trips to pick stuff up as a sole trader.... It's not earning you money doing the trip, the item is not yet commissioned into your fleet and it would be unviable to run a tachograph for the journey once a year..... Opt for common sense approach and IF it becomes a problem be prepared to argue in court..... FYI I stop for breaks in that regard, take time stamped photos and write it down when needed.

Whilst tachographs are a "legal requirement" the key laws are around EU drivers hours breaks.