I was able to do this. Here are some steps and other notes for anybody who is curious:

• Restore Content to Local First, I restored all content from "Account A: Project A: Live" to my local.
• Update Config Files I updated the Umbraco.Deploy.ApiKey app setting in my web.config and I updated the environment info in the ~/Config/UmbracoDeploy.config file.
• Increase Timeouts I increased the timeout in the ~/Config/UmbracoDeploy.Settings.config file to 30 minutes (see code snippet below).
• Deploy Files I deployed the files from the old Umbraco Cloud account/project to the new Umbraco Cloud account/project.
• Upgrade Umbraco There was an Umbraco version mismatch (between the files and the database). For now, I just ran the upgrade on Umbraco Cloud, but that probably borked something, so when I do this for real I'll probably sync the Umbraco version before deploying the files.
• Fix Media Items I kept running into an issue about media files not existing. Turns out the umbracoDeployDependency and umbracoDeploySignature tables had incorrect data for some reason. I got around this by saving all media nodes again (see code snippet below), and then by saving another media node that the automated process seemed to miss (the error message indicates the media node ID, which helps to locate it).
• Transfer Local to Live I was finally able to queue a transfer that ran successfully (after a number of failed attempts). This was from my local to the "Account B: Project B: Live" environment.

Thanks again, Paul, for your tips.

Code snippet for UmbracoDeploy.Settings.config:

<?xml version="1.0" encoding="utf-8"?>
<settings xmlns="urn:umbracodeploy-settings">
  <deploy sessionTimeout="1800" sourceDeployTimeout="1800" httpClientTimeout="1800"/>
</settings>

Code snippet to resave all media nodes:

var allMedia = Umbraco.TypedMediaAtRoot().SelectMany(x => x.DescendantsOrSelf())
    .Select(x => x.GetKey())
    // Need to process less than 2,000 at a time.
    .Take(1500)
    .ToArray();
var mediaService = ApplicationContext.Services.MediaService;
var mediaNodes = mediaService.GetByIds(allMedia);
foreach (var item in mediaNodes)
{
    mediaService.Save(item);
}

Summary: I was able to transfer the content from one Umbraco Cloud project on one account to a different Umbraco Cloud project on a different account.