Does Israel's Temple have to be functioning for the entire first half of the seven years?
No. The context indicates Israel's Temple only needs to be in operation by the midpoint of the seven-year period, because the sacrifice and oblation must cease then. Whether it functions for the entire first three and a half years is beside the point; what matters is that Temple worship is active by that midpoint.
Les teaches that the seven-year treaty triggers God's time clock back into gear, and then seven literal years unfold before Christ returns. The language about the middle of the week says He will cause the sacrifice and oblation to cease, meaning Temple worship stops. For Temple worship to stop, it must first be in operation. So Israel will rebuild her Temple and have it operating by the midpoint. Whether she has it the whole three and a half years is beside the point. The context also notes the Antichrist likely gives Israel permission for Temple worship as part of the covenant.