The height being pumped to is called the "head", this is listed in the pump specifications and they usually include a table of how quickly the pump can move water at various head heights. A quick look at basic condensate pumps on the Home Depot website showed them with useful ratings up to 10 feet of head. The pump will move water more slowly to a greater head, but that will be no problem for running periodically to clear condensate water.
The Midea Cube 50 Pint with pump is one of the most efficient dehumidifiers on the market and has a built-in pump that can move water up to 14 feet high. If your current model is older or less efficient it might make more sense to upgrade rather than just buying a new pump.
The most important thing is efficiency! Dehumidifiers run a lot and so the electricity you pay for over time can often be 2-3X the cost of the unit. It's worth it to pay a bit more for one that is highly efficient. This is the one I would recommend: https://www.amazon.com/dp/B091BYVD2W
The Midea Cube 50pint dehumidifier is one of the most efficient units and has a pretty convenient design. They also sell versions with a built-in pump if you don't have a drain next to it. The pump model is currently sold out on Amazon but you can find it at other stores. Here are the EnergyStar.gov ratings for dehumidifier efficiency.
If you have mold growing you have a problem to fix, but this company doesn't sound like they're trustworthy to tell you what to do rather than just trying to get you to spend money. The right fix will depend on the construction of your basement and how much mold there is.
If you have a small amount of mold growth and the basement is relatively sealed you might just need a dehumidifier and to spray the affected area with [Concrobium mold control spray](https://www.amazon.com/Concrobium-25326CAL-Mold-Control-Spray/dp/B084CNTQX2/. This is a non-toxic, mineral based product which works by forming tiny crystals as it dries that literally shred the mold cells.
If the mold growth is more severe you might need to replace the affected wood or drywall or whatever. If your basement is currently vented to the outside or you have exposed dirt the fix might be more involved. Nobody should be suggesting a need to spray disinfectants everywhere or anything like that.
Look at the Midea Cube 50 pint with pump. It's one of the most energy efficient models currently available (Energy Star efficiency ratings are here) because it uses the latest generation of refrigeration loop. It also has a built-in pump so you can drain to the outside, and the overall design is pretty clever and stores compactly. The government changed how dehumidifiers are rated to be more realistic, 50 pints per day under the new system is equivalent to 70 under the old one.
The main benefit of commercial dehumidifiers is that they can get wet spaces dry much more quickly, which is important for restoration. If you just need to maintain a space at reasonable humidity a regular dehumidifier is probably plenty for your needs.
Check out the Midea cube 50 pint with pump which is currently on sale at Amazon. This is one of the most efficient dehumidifiers per Energy Star testing, and a pump so you don't have to manually empty the tank is nice. The overall design is also rather clever and compact. Note that the federal government recently changed how dehumidifiers are rated to be more realistic, so what we call 50 pint models today used to be called 70 pint models.
The actual core refrigeration loops in dehumidifiers all come from the same factory in China so there shouldn't be significant performance or reliability difference between brands. The efficiency differences are mostly due to what generation of the loop is included, with the most efficient using the newest and most expensive version. This is why when there was that big dehumidifier recall a few years back essentially every brand and model were included.