You must apply the higher stat, no doubling of PC stat
Your game statistics are replaced by the statistics of the beast, but you retain your alignment, personality, and Intelligence, Wisdom, and Charisma scores. You also retain all of your skill and saving throw proficiencies, in addition to gaining those of the creature. If the creature has the same proficiency as you and the bonus in its stat block is higher than yours, use the creature's bonus instead of yours.
For a particular skill, there are two proficiency bonuses contemplated in the above text: your humanoid form proficiency bonus, and the beast stat block's proficiency bonus. You propose a third category: a hybrid proficiency bonus with the PC's proficiency as base, augmented by the beast's 2x proficiency feature.
When in doubt, the rules of the game only do what they say they do. Every sentence of this text involves a direction to pick between either a PC number or a Beast number. There is no mention of calculating new numbers. Instead, the text states that you must choose the "bonus" that is higher. Here, "bonus" probably refers to "skill bonus", as defined in MM chapter 8: "A skill bonus is the sum of a monster's relevant ability modifier and its proficiency bonus". The text dictates that you must use the higher of two bonuses: you may not invent a third number.
MM Chapter 8 continues:
A skill bonus is the sum of a monster's relevant ability modifier and
its proficiency bonus, which is determined by the monster's challenge
rating (as shown in the Proficiency Bonus by Challenge Rating table).
Other modifiers might apply. For instance, a monster might have a
larger-than-expected bonus (usually double its proficiency bonus) to
account for its heightened expertise.
The saber-tooth does not have an ability that doubles its proficiency bonus. There is no evidence that "heightened expertise" is anything other than flavorful description. If "heightened expertise" were a real ability, it would appear in the monster's statblack. Again, when in doubt, the rules only do what they say they do. From the MM, all we really know is that a monster has a higher proficiency bonus in a particular skill merely because it says so.
Hit point modifiers are likewise determined arbitrarily. Bugbears and Copper Dragon Wyrmlings are the same CR, with the same Constitution, but the Wyrmling has +4 and the Bugbear has +5. That's just how it is.
Because the higher proficiency bonus is not tied to an ability, you have no basis on which to apply said nonexistant ability to a proficiency bonus imported from your druid.
The 5e Bounded Accuracy system is meant to reduce the amount of math that a player has to do on the fly. This is in contrast to Pathfinder, where a druids may resort to excel spreadsheets to calculate how different modifiers and buffs apply for all of their various beast shapes.
DMG Chapter 7's section on Proficiency Bonus does not speak explicitly to your question, but it emphasizes the focus on picking a single number and applying it once:
Occasionally, your proficiency bonus might be multiplied or divided (doubled or halved, for example) before you apply it. For example, the rogue’s Expertise feature doubles the proficiency bonus for certain ability checks. If a circumstance suggests that your proficiency bonus applies more than once to the same roll, you still add it only once and multiply or divide it only once.
Note that, in the above text, a doubled proficiency bonus is not treated as a proficiency bonus applied twice: the doubled proficiency bonus is the new proficiency bonus.