How do I manage periods for abbreviations with parentheses, right before a period?

If you are only concerned with this example you could just not use the abbreviation and write:

... information (blood test, x-ray, et cetera).

You are correct in this item from your example:

(blood test, x-ray, etc.).

The two marks of punctuation are actually different; the period following "etc" marks an abbreviation, and the period following the closing parenthesis is terminal punctuation. Each mark is doing a different job, so you need both of them.

Here's another example that arises frequently:

The tarantula crawled into the sleeping bag ... unnoticed.

In this case, the three periods are marks of ellipsis, here serving to create a dramatic pause. But, when the marks of ellipsis fall at the end of a sentence, you still require closing punctuation:

The surgeon raised the scalpel and leaned forward ....

Since c is not the last letter of cetera, you need a full stop after it (at least in formal situations). British usage is to put this before the bracket in all cases, though I believe American differs sometimes. And though, logically, you would need another full stop to end the sentence, I've never seen it used (probably to avoid ending with ?etc..)

So go with the third alternative, "...x-ray, etc.)"