Collecting sperm during relations with a sterile, unlubricated condom is one halachically acceptable method of collection, though it is prohibited during niddah. Please see here for more discussion of the alternatives.
We follow the viewpoint that IUI (intrauterine insemination) is permissible during niddah when necessary. There is a range of opinion on how the procedure affects counting the clean days. Please see our discussion here, and feel free to get back to us with more details about the timing of the procedure within your cycle if this becomes relevant.
Supervision of the sperm until it is inserted can help to ensure that there is no mix-up. While a number of halachic authorities strongly prefer that all procedures be conducted with supervision, others rely on the existing safety precautions of reputable clinics.
As you note, there are a lot of steps involved. We recommend arranging a free consultation with a Yoetzet Halacha Fertility Counselor, details here, to help you navigate the halachic issues involved. We hope your procedure is smooth and successful.
Please let us know if we can be of further assistance.
This response was updated on 2 May, 2021.