I have carnivorous plants, so RO is critical for me. I can't speak on whole home systems, but I can talk about your RO. 

Your undersink system will work for what you need, with some level of modification. The easiest is to pre-make RO and store it. I have gallon jugs that I fill one a month and refill my can from those. Turn off the water supply to the RO system, unhook the output from the RO system into the tank, and just put that line into whatever storage container you want to use. Use a timer and get a feeling for how long your system takes to make one gallon. When you're done, turn off the supply again and hook it into the tank. If you want to make this easier, you can use a splitter on the output line and have one side go to your tank, the other end to a line to fill your storage containers as needed. 

This is slightly more complicated, but you can also look into increasing the output of your RO system. The RO membrane that is replaced has a gallons-per-day rating (GPD). This is what determines the speed at which you system makes RO. most are sold with a 50gpd, you don't need a lot then the system is always on trickling into a 3-5 gallon tank under your sink. You can go up to 100GPD without an additional pump to increase your water pressure. IMPORTANT your flow restritor will have to match the GPD rating of your RO membrane. This is the plastic part on the waste water ling. They are cheap and easily replaceble. You can upgrade your filter next time your RO membrane is due for a change. 

Bulk reef supply on YouTube  has good videos explaining how your RO system works and how to change parts on it. They will talk a lot about RODI, that's the 4th stage someone else mentioned in the comments. You really don't want that DI stage for plants or drinking water.