The issue is not whether the item may be carried on, but whether it is of a size consistent with the poorly-written rule which would be far better if it stated the exact maximum dimensions.
As it is written and if it is strictly interpreted, UA's carry-on size limits are published to FAA and then become enforceable as FAA rules. OP says that the unit is "slightly" larger than those limits and will require a case beyond that. Will it be too large under the rules.
If the bag is within size limits, also pay attention to the size of the aircraft. OP says he is going to Florida, but does not say from where. On smaller aircraft, the bag may simply not fit.
Contrary to advice above, I would contact UA and ask to speak with the "Accessibility Desk." Have the exact dimensions of the item in its case. Include handles, wheels and whatever else might protrude. If the item will fit and is within UA's rules, ask that the appropriate entry be made in your PNR notes.