It will help greatly if the OP provides information on the exact make and model heater they bought.
The 110V tankless is designed as an undercounter instant water heater for use with sinks. Many stores and businesses use them to supply their restroom sinks. If the incoming "cold" water is not bitter cold, it might work if you run the shower VERY slow (I mean like a dribble), but cold water in the wintertime is never going to get more than luke warm if that.
What would work (I know because I did it as a temporary installation for a friend while I re-plumbed her 150 year old house) is a instant electric heater that is 240v powered and draws 11,000 or 13,000 watts . It wasn't far from her electric panel so I did a drop of 6 gauge wire to the crawl space and 15 ft or so to the water heater mounted on a board screwed to a floor joist. Eventually all the plumbing was finished and I installed a gas Noritz instant water heater and removed the temporary electric one.
Find product information, documents, videos, and installation assistance for the Rheem Professional Classic Series: RTEX-08, RTEX-11, RTEX-13
Now, look at the chart about half way down the first page on the left side of the flyer below.
https://s3.amazonaws.com/WebPartners/ProductDocuments/3320FFD8-8A08-46F9-AF9F-04AFEDF86EEC.pdf
It is the critical information you need to know. You want water at about 100°F or slightly more, and a shower head with the restrictor installed is supposed to flow no more than 2.5 gallons per minutes. Using the chart and looking at the model RETX-11 we see that at 2.5 gpm this heater will raise the temperature of the water no more than 30°F. This means that wintertime water coming in that is COLD, say 50°F will never get more than 80°F max, which is a lukewarm to almost chilly shower.
I use the example of the RETX-11 because it is 11,000 watts, at 240v is 46 amps draw. You can install it on a 60 amp breaker and meet the electric code (you cannot exceed 80% of circuit capacity with heating devices). You can go to the RETX-13 and gain a few degrees of temp rise, but at 54 amps you are past the 80% limit on the circuit (and NEC does not allow more than a 60 amp circuit per heat element, which is why you find multiple breakers on your home back up heat strips or on a whole house electric instant water heaters).
Bear in mind, you will have to run a completely new circuit from the panel to the trailer using 6 gauge wire. (for the heater I suggest) This in itself can be quite expensive, as wire has skyrocketed in price.
For your "110v" water heater, if it is one of the ones designed to plug into a standard outlet, it will barely heat water enough to wash your hands. The smallest one in the chart I reference above is 29 amps which will require a 40 amp breaker, and depending on the distance you can get away with 10 gauge wire or possibly step up to 8 gauge. You are actually better off with a 240v water heater as it will put out more heat and use smaller wire doing it, thus saving money.
Charles