"There is no "house" there of any kind. "House" is "oikos"
I think you will find the word house is in verse 3:4, 3:5, 3:12 and in 3:15, and in fact, "which is the church of the living God, the pillar and ground of the truth", modifies it. And you are right the "word there is "ekklesia", that relates to community or assembly", not an institution or place. Since Paul is using the word house in 4,5, and 12 to mean family or household, it stands to reason it carries the same meaning in 15.
1Ti 3:15 "But if I tarry long, that thou mayest know how thou oughtest to behave thyself in the house of God,which is the church of the living God, the pillar and ground of the truth."
"Since Paul is using the word house in 4,5, and 12 to mean family or household, it stands to reason it carries the same meaning in 15."
The word for family is close; its oikogennia.
You are right, "oikos" it is in the context, but not in the disputed passage. Definitely, the family of God or the household of God are valid descriptions of the historical Church. Even the plan of salvation is called sometimes economy (oikonomia) of salvation. This is a powerful scripture to contemplate when thinking of the Communion of Saints.