SHBG is not time of day dependent. You can get it tested at any time. LH/FSH ate time dependent and should be tested first thing in the AM. So should T if you are not on TRT. If you are on TRT, then it doesn't matter what time of day, simply test just prior to the next scheduled injection.
HCG should be split into a minimum of 3X per week. Many guys do it on a MWF schedule. Simply divide 1000/3 = ~330-340 IU per injection If you are on a E3D TRT schedule, then on the days of T injection is a good sustainable protocol. In this case, 1000/7 X 3 = ~430 IU per injection.