Base tax is a factor in Vassalising a nation in the first place. So if a nation has a relatively high base tax it will have a negative effect on your attempts to Vassalise them.
This can vary from +30 to -100 and is calculated by:
60*(your base tax)/(target's base tax)^2 - 90
If the nation is already your Vassal which from what you describe, they are, then the base tax shouldn't be a problem. It shouldn't factor in to whether the Vassal is disloyal or not. I think this instead relies on similar factors to say a Personal Union claiming independence.
I.e. Low relations with the vassal or low prestige
Edit to include info from comments below:
Other factors that effect a nations ability to Diplo-Vassalise another nation (but do not effect a Vassals opinion of their Overlord)
- Negative opinion modifier if their army is larger than your own
- You can not vassalise a nation who's base tax is higher than 40
This is actually a confusing issue because the DLC expansions wildly change the mechanics for Terra Incognita, and there is very little documentation about this on the wiki.
'Vanilla' EU4 Terra Incognita Discovery Mechanics
In 'vanilla' EU4, you discover new ocean provinces by moving a fleet with an Explorer on to the unknown ocean tiles. You also randomly reveal coastal land tiles when you move into an adjacent ocean province. To normally reveal land tiles, use an army with a Conquistador. An army with an a Conquistador can also directly disembark from a fleet containing transport ships to a coastal province with Terra Incognita.
DLC Expansion EU4 Terra Incognita Discovery Mechanics
If you have certain expansions enabled (El Dorado), the mechanics are almost completely different. Explorers can no longer move into Terra Incognita tiles at all. Instead, any group of 3 or more Light Ships or Transports will have a "Send Explorer" option which shows you a list of possible 'Exploration Missions'.
There are two different types of Exploration Missions. One is "Explore the waters of X", which sends the fleet out to systematically reveal ALL ocean provinces in the 'X' region. The second type of mission is only available after the first Exploration Mission is completed (or partially completed), and it is "Explore the coast of X", which sends the fleet out to systematically reveal ALL coastal provinces in the 'X' region. Which missions are available to you depends on your Colonial Range from one of your provinces to the region in question. You need a certain range to enable the first mission and a bit higher range to enable the second type of mission.
Comparisons and Conclusions
There are pros and cons to each approach. With vanilla mechanics, there is no limit to how far you could reveal Terra Incognita. You could take your very first fleet directly from Japan to Africa and disembark and conquer the Kongo. With the DLC expansion mechanics, this is impossible. However, you suffer no naval attrition damage when on an Exploration Mission. Some missions, like "Explore the waters of Australia" or "Explore the waters of Oceania" would take many, many trips for an Explorer to complete without the DLC expansion mechanics enabled. Additionally, unless you are using the disembark method, it can sometimes be hard to reveal the coastal land province you want using the vanilla mechanics, since the provinces you reveal are completely random (and not remotely comprehensive).
In my opinion, neither version is superior. I play with the DLC expansions enabled because the other mechanics they provide are excellent. However, if you were trying to do some very specific early exploring (for the purposes of pure conquest), I could see how the Exploration Mission mechanics would be limiting, since they are based on the premise that you would want to colonize the regions in question.
A Note About Full World Exploration
Note that there is no way (with either mechanic) to reliably discover uninhabitable land patches of Terra Incognita. You have a chance to discover them over time if you have a province near the area (or an ally does), but it's pure luck. It makes getting the World Exploration achievement annoying when you have 99.9% of the world explored and are only missing a patch of uninhabitable land in Siberia.
Best Answer
Getting rid of the PU
There are several ways and you should get rid of the PU asap as the Frances strength will also fuel liberty desire in your colonies:
Incorporation of the junior partner into the senior partner, by integration or inheritance
The junior partner declaring independence and winning the war
Declare any war and surrender with release France.
The ruler dying while the junior partner has a negative opinion of the overlord
Pretender rebels enforcing their demands in the junior partner
Rebels enforcing demands that change government type
Peacefully abandoning a PU with a lesser partner
So, my recommendation owuld be to simply wait for the independence war and surrender it or declare war on anybody supporting France independence and surrender to them while releasing France. In any case, after oyu release France, you will have a truce and won't be able to ally them during that truce.
Getting France to help
Not possible beyond deliberately luring enemy armies into french stacks. When a subject has over 50% liberty desire, they stop helping their overlord.