No, Double Strike is its own standard action. It cannot be combined with any other action of any kind. Only non-action effects (e.g. Power Attack) can be combined with it.
Other actions can have effects that last, and can affect future actions like Double Strike—for example, true strike takes its own standard action, and applies a +20 bonus to your next attack. If your next action is the first attack in a Double Strike action, then that attack gets the +20 bonus from true strike. This is totally different from what you are describing, however: you are not mixing the actions together at all. You are just benefiting from some temporary bonus provided by true strike.
This is because actions themselves never happen at the same time. If two different things say “As a (not-free)1 action, you do this,” and there is no lingering effect or bonus from either, those two things can never benefit from one another. So when Double Strike says “As a standard action, you can make one attack with both your primary and secondary weapons (or with both ends of a double weapon),” and full-attack says “you must use a full-round action to get your additional attacks,” you know those two things can never, ever, happen at the same time. The rules, effects, bonuses, penalties, and so on from each never affects the other in any way.
So if you are using a standard action for Double Strike, you are not using a full-round action for a full attack, by definition. Likewise, if you are using a full-round action for a full attack, you are not using a standard action for Double Strike, again by definition. There is absolutely no way around this. Even if you somehow had more actions in a single turn, you would have to do one first, then the other, not both at once. Any exception would have to be completely explicit, something like
When you make a full attack with the two-weapon fighting option, your first attack can be replaced by the two attacks Double Strike usually performs as a standard action. If you do this, you [figure out how the penalties work because it’s spelled out for you in this ability that lets you do this thing you normally could not do].
Note that no such effect actually exists in the game at this time. But you can see a similar example in the initial benefits of the Greatsword Battler divine fighting technique, which allows you to use Vital Strike (normally only applying to the particular standard action known as Attack) at the end of a charge (usually a full-round action independent of the Attack action).
- Some free actions are explicitly allowed to occur during some other types of actions. For example, talking is allowed pretty much no matter what you’re doing. Five-foot steps are allowed between attacks in a full attack. Etc. Such exceptions are always explicitly described by the rules for the free action in question. Non-free actions never have these kinds of exceptions. Ultimately,
these exceptions are not particularly relevant to this question.