Removing the lead at the base would introduce decay to the base of the tree, right? You need a waiver from them to do that and I would not do it since the remaining leader seems to have established lean in the same direction. In ten years your client could have a mess on their hands.
Reduction pruning on the leader closest to the house is good no matter what else you do. Do you have the ANSI A300 Part 1 Pruning standard and the accompanying Best Management Practices booklet? Those will likely lead you to write a specification to make 3-5 cuts in
It would be great to chip the branches and mulch the tree with the chips - 3" thick of mulch, with the mulch 3-6 inches removed from the root flair itself. Here is a link to literature describing the benefits of mulching with fresh arborist chips:
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puyallup.wsu.edu
You may want to vet your client for long-term commitment capacity. If they seem good for it, consider a synthetic polymer cable, which lasts for about a decade on my brand label or until the tree outgrows it. The specifications for Branch Saver include a 30 inch bury after you've spliced each side with a teardrop shape, so you need ~80 inches of distance between those two leaders. There are other synthetic polymer cabling materials on the market - Yale makes one and so does Notch. Their specifications are each different and you may want to choose the cabling material based off of the specification that makes the most sense to you. If the clients seem like they might forget, die, or move, talk to them about metal cabling instead. Also look into metal cabling if there is not enough room to install synthetic polymer cabling. 5/16ths EHS metal cable is a pretty common size/material, but some guys prefer the non-ehs that they can splice. There is not a consensus about the best way to terminate metal cables. I use Endz terminations with EHS. Rig Guy and Wedgegrips are other options to use with EHS. Do you have the ANSI A300 Part 3 supplemental support systems standard and the accompanying Best Management Practices booklet? Those will protect you from liability and your client/client's tree from a bad installation. Make sure that you include the phrase "to provide additional support" in your specification to indicate that the cable is part of the structural system and does not guarantee against failure even if properly installed. Make sure you know whether you need to pretension your system and how much. The word "taut" sometimes comes up in the ANSI standard, particularly for all metal cable installations which are intrinsically weak in the dimension of shock loading and strong in terms of tensile strength. The manufacturer of synthetic polymer cable may have a "tautness" clause specific to their product that is or is not at odds with your synthetic cabling objective.
You mention that there is no bark inclusion at the base - good work noting that. There will be an inclusion as the diameters of the two trunks increase. Additionally, the aspect ratio is okay - could be better, could be worse. It is easy and thus cost effective to install bracing rods at the base using that ANSI A300 Part 3 to guide the way. I do not have an opinion about whether bracing rods should be installed. It may be good to communicate the benefits and drawbacks to your client about installing bracing rods so that they can make that decision. Or, you may feel like it is not worth mentioning. You'll want a big drill - probaby a superhawg that you can borrow from someone until you can buy your own. Be careful about using a big drill without a clutch, e.g. a hole hawg - you can injure your thumb. A corded big drill needs a very large gauge cord, so try to turf up a lithium drill.
hth, you are starting out with a good mindset.