If I employed three operators to carry out three distinct (but inter-related) functions with different specialist tools, then I would need to set up protocols on who does what and how they transferred work between them in order to ensure the whole operation ran smoothly.
You need to think in exactly the same way. Begin by making a list of the particular reasons for using each tool (as you have begun to do in your question) and then ensure that NOTHING ELSE is done in that tool.
Make a list of which jobs are to be done in which package and pin it to the wall beside your computer. Because I like to think in pictures, I would draw a diagram showing information moving between tools as different tasks are carried out by each.
In particular, you must select one (and only one) program for data entry and be strict about NOT "updating each program depending which I have open". That is a recipe for disaster. When all the new or updated data has been entered through your gateway application, then you can transfer files to each of the other tools.
You need to identify the file type that is needed for import into each tool and then determine which other software can produce that type. It may be which package you use for data entry will be decided by the fact that it can export in forms suitable for each of the others.
Be prepared for the possibility that none of three you now use will be suited to the data entry task under these strict rules. Do not worry that selecting a fourth tool will make matters worse. If you choose correctly, using four tools properly can be more efficient than trying to get by using three poorly. (Of course, making the wrong choice will cause you to try to bend the rules posted on the wall to work around deficiencies in the software, and that will make things worse!)