The resource share amounts I have set in BAM for projects does not seem to be updating on my client. I have synced with BAM and other changes (such as those to work preference) update as expected.

The place I've been changing the resource share is the "Projects" page on boincstats.com. https://www.boincstats.com/bam/projects/
Should I be changing it on the hosts page instead? https://www.boincstats.com/bam/hosts/

You can do both, but the 'hosts' setting will override the 'projects' setting.
If you want to have a different resource share for each host, the 'hosts' page is the only way to go.
If you want to have the same resource share for all hosts on a project, make sure that for all hosts (using the 'hosts' page) the resource share is set to -1 (displayed in the list as "Project default&quot. Then the resource share of the 'projects' page should be applied on the client.

If you want to configure resource share on the project's webpage, both the 'hosts' and 'projects' on BAM! should be set to -1 (Project default)

All of the above obviously implies that you have attached the projects to your client using BAM! and not manually added...

Did you by any chance divide your hosts into host groups? Because in that case, you would also need to look at this page to configure resource share: https://www.boincstats.com/bam/hostGroups/
Same principle applies - hostgroups override project settings - set hostgroups to -1 to use project instead.

That was my issue
For some reason the host settings were set to not use project default. I don't know if they changed on their own or if I'm just a fool that can't remember changing them.

Somewhat related, BOINC isn't letting resource shares be set above 1,000. BAM let's you set it up to 10,000, but the BOINC client just sets anything above 1,000 down to a cap of 1,000.

Upon further investigation, the amounts don't update (at least visually) when syncing with BAM. However, if I use the "Update" command on a project it will correctly update.

Are you certain that you haven't set both 'hosts' and 'projects' resource share to -1 (default)? Because in that case, the resource share is defined by the project preferences (on the project site - not on BAM!). And then indeed the value will only update when the project is updated.

That's what I'm currently doing. If I want resource share to ONLY be decided by the Projects page, how should I set it up?

Set all 'hosts' and/or 'hostgroups' resource shares to -1 (default). Projects page setting to whatever you want.
But don't click on the 'show hosts' icons to set share, because that will be the same as using the 'hosts' page.

Click the img links below to see what it should look like.

Projects: https://ibb.co/HD4vR6GY
Hosts: https://ibb.co/xTQ3xCh

That's what I have it set to. I should be good from here.