Is working with two clients, whom I met through the company I co-founded with my partner, outside the framework of this company and without his knowledge, after working hours and using my own tools, considered Haram and a betrayal of the partner, and does the Hadith: "Sin is that which wavers in your soul and which you dislike people finding out about" apply to it?

If the work is outside the company's official hours, using your own tools, and you did not solicit the two clients but they accepted your offer and refused to work with the company, then there is no objection to accepting that work and its compensation for yourself, provided that you do not exploit the company or its name, and that it does not violate the partnership agreement. The hadith "Sin is what wavers in your chest and you dislike for people to know about it" does not apply to you, for not everything one dislikes people to know about is a sin; it may be permissible or lawful. The hadith applies to one who is doubtful about a matter, not knowing whether it is a sin or not. Sin is that which causes unease and constriction in the chest, and which people disapprove of when they learn about it. As for matters for which there is a textual religious ruling, the duty is to obey Allah and His Messenger and submit to it, even if one's heart is not at ease with it; that has no bearing.