You must apply the highest statistics, without doubling the PC statistics
The statistics of your game are replaced by the statistics of the beast, but you retain your alignment, personality and intelligence scores, wisdom and charisma. He also retains all of his saving abilities and shooting abilities, in addition to obtaining those of the creature. If the creature has the same ability as you and the bonus in its statistics block is higher than yours, use the creature's bonus instead of yours.
For a particular skill, there are two competition bonuses contemplated in the previous text: its humanoid ability skill bonus and the beast statistics block skill bonus. You propose a third category: a hybrid competition bonus with the ability of the PC as a base, augmented by the beast's 2x competition function.
When in doubt, the rules of the game only do what they say they do. Each sentence in this text implies an address to choose between a PC number or a Beast number. The calculation of new numbers is not mentioned. Instead, the text indicates that you should choose the "bonus" that is higher. Here, "bonus" probably refers to "skill bonus," as defined in Chapter 8 of MM: "A skill bonus is the sum of the relevant skill modifier of a monster and its competition bonus." The text dictates that you must use the greater of two bonds: you cannot invent a third number.
Chapter 8 of MM continues:
A skill bonus is the sum of the relevant skill modifier of a monster and
your competition bonus, which is determined by the monster's challenge
qualification (as shown in the table Competition bonus by challenge rating).
Other modifiers can be applied. For example, a monster could have a
greater than expected bonus (usually double your competition bonus) to
Account of your greatest experience.
The saber tooth does not have a skill that doubles its competition bonus. There is no evidence that "more experience" is anything other than a tasty description. If "enhanced experience" were a real skill, it would appear in the monster's statblack. Again, in case of doubt, the rules only do what they say they do. From the MM, all we really know is that a monster has a higher competition bonus on a particular skill simply because it says so.
Hit point modifiers are also determined arbitrarily. Bugbears and Copper Dragon Wyrmlings are the same CR, with the same Constitution, but the Wyrmling is +4 and the Bugbear is +5. This is how it is.
Because the most competitive bonus is not linked to a skill, you have no basis to apply that nonexistent skill to a competition bonus imported from your druid.
The 5e Bounded Accuracy system is intended to reduce the amount of math that a player has to do on the fly. This is in contrast to Pathfinder, where druids can turn to Excel spreadsheets to calculate how different modifiers and benefits are applied to all their various forms of beasts.
The DMG Chapter 7 section on Competition Bonus does not explicitly speak to your question, but emphasizes the focus on choosing a single number and applying it once:
Occasionally, your competition bonus can be multiplied or divided (double or halved, for example) before applying it. For example, the rogue's Expertise function doubles the proficiency bonus for certain skill checks. If a circumstance suggests that your competition bonus is applied more than once to the same roll, you still add it only once and multiply or divide it only once.
Note that, in the previous text, a duplicate competition bonus is no treated as a competition bonus applied twice: the duplicate competition bonus is the new competition bonus.