Upvote:2
Embassy can't hurt
You mention that the German Embassy said they cannot help, however you don't specify what you asked them specifically. While I don't think they would "backstamp" your passport, it should be pretty easy for them to provide you with an official document stating that Kmi with passport number 1234 is in front of them in the Embassy located in India at date XYZ, despite passport 1234 not bearing an exit stamp, and they produce such document at the passport holder request.
I assume more than your allowed haven't passed since you arrived to Schengen area, so it doesn't really matter the exact day you left, as if you are in India today you cannot have overstayed.
While not really required (as mentioned by other answers), such document should calm your sponsor or, really, any issue in the future where the lack of exit stamp could be questioned.
Upvote:6
The Schengen Borders Code states:
“The travel documents of non-EU nationals are systematically stamped upon entry and exit. If a travel document does not bear an entry stamp, it may be presumed that the holder does not fulfil, or no longer fulfils, the conditions of duration of a short stay. However, the non-EU-country national may provide any credible evidence of having respected the conditions relating to the duration of a short stay, such as transport tickets or proof of his/her presence outside the territory of the EU countries. On the request of a non-EU-country national, the insertion of an entry or exit stamp may be dispensed with if this insertion is liable to cause difficulties for the person. Instead, the stamp must be recorded on a separate sheet indicating the person’s name and passport number.”
Source: https://eur-lex.europa.eu/EN/legal-content/summary/schengen-borders-code.html
Keep evidence of the date you exited Schengen (eg boarding pass, proof of arrival in India etc) and you and your sponsor should not have any problems in future.