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You can post questions, and if you know the answer, you can also post answers.
To your question.
The name is probably not "Dr. John Smith senior Foundation" in German, so you half translated it already. You might as well use English orthography for "senior" as well. This is "Snr" or "snr" or "Sr" or "sr", and another 4 variants with periods following the "r".
A search of Google for "George Bush Snr/Sr" gives:
325,000 hits for Sr, and only
26,700 for Snr.
The use of the period is not necessary, since "r" is the final letter, but it seems to be used in most cases.
Thus, the popular translation would be "Dr. John Smith Sr. Foundation", as your stomach has already told you.
(Others might disagree).
What audience is your translation for? - that might be a factor in whether you translate it at all.