I figured out a solution. Documenting here for anyone else who comes across this and needs to know:
1. Install boroda74's
Additional Tagging and Reporting Tools. (Be sure to download the latest from the dropbox link, as there may be earlier versions posted directly on the page.)
2. For ease of these operations, make sure Column Browser is turned on.
3. Follow Steven's instructions above to ensure your old MediaMonkey tags are mapped correctly in MusicBee. Now, if you don't like the messy old MediaMonkey tag names, then you want to create new tags and migrate the old values.
4. In Preferences > Tags (1) > Define New Tags... , create a new custom tag row to match each old MediaMonkey tag row. (So if you have 3 old MediaMonkey tags listed there, add 3 new custom tags with new names to match them.)
5. Right-click the Column Browser and Set Displayed Columns to include the old and new custom tags you're going to work with. In the left sidebar, choose the scope of what files you want to work with (folders, albums, etc.).
6. Now, let's say you have an old custom tag "MediaMonkey Occasion" and a new custom tag "Occasion." In the column browser under the MediaMonkey Occasion column you've displayed in (5) above, shift-select all the tag values represented there. MusicBee will display all of the tagged tracks in the Main Panel. Select all of those files.
7. Choose Tools > Additional Tagging Tools > Copy tag... and a panel window will open. For the "Copy tag" dropdown, in this scenario you would select "MediaMonkey Occasion" and for the "to tag" dropdown you would select "Occasion." (Your actual tags will vary per your actual files' tags.) You may want to Preview to see the effects, and then click Proceed once you're sure the results are what you want. MusicBee will copy the tags from the old tag field to the new tag field. Close the Additional Tagging Tools window.
8. You'll see that all the pertinent tag values are now shown under both the "MediaMonkey Occasion" and "Occasion" columns. You may want to manually click on the values and check the filtered results in the main panel just to doublecheck that the tags copied correctly.
9. Now it's time to get rid of the old MediaMonkey tag field. First, select all the Occasion tag values in the Column Browser, then select all the files shown as filtered in the main panel. Right-click to Edit the files. On the "Tags (2)" tab, check the box for the old tag (in this case "MediaMonkey Occasion") and leave the value blank. Click the Save button. MusicBee will now update the files to remove the old MediaMonkey tags.
10. Repeat steps 6-9 for each MediaMonkey/MusicBee custom tag pair until you've converted all the old MediaMonkey tags to MusicBee tags.
11. Now it's time to clean up. In Preferences > Tags (1) > Define New Tags..., click the minus (-) signs to remove all the old MediaMonkey tags. (But be sure to leave the new MusicBee tags alone!) Click Save and Close; then, in the custom tags list, revert any unused tags back to the default "Custom3", "Custom4", etc. values like they were when MusicBee was installed. (But again, leave all the new MusicBee tags alone!) For each of the reverted tags, ensure "Save to MusicBee database only" is selected. Apply, Save and Close.
12. In the Column Browser, Set Displayed Columns to remove the old (now deleted) MediaMonkey tags.
And that should be it. Now you've got all your old MediaMonkey tag values in natively-named MusicBee tag fields.