I had a similar problem trying to get Chrome to be a default browser in a sandbox. After hunting around for about 4 days, I think I have it sussed and have the below .reg file which does the trick and makes Chrome show up in the default browsers popup in settings. you should be able to adapt for your needs.
Windows Registry Editor Version 5.00
[HKEY_CLASSES_ROOT\ChromeHTML]
@="Chrome HTML Document"
"AppUserModelId"="Chrome"
[HKEY_CLASSES_ROOT\ChromeHTML\Application]
"ApplicationCompany"="Google Inc."
"ApplicationDescription"="Access the Internet"
"ApplicationIcon"="D:\\Sandbox\\Chrome\\drive\\C\\Program Files (x86)\\Google\\Chrome\\Application\\chrome.exe,0"
"ApplicationName"="Google Chrome"
"AppUserModelId"="Chrome"
[HKEY_CLASSES_ROOT\ChromeHTML\DefaultIcon]
@="D:\\Sandbox\\Chrome\\drive\\C\\Program Files (x86)\\Google\\Chrome\\Application\\chrome.exe,0"
[HKEY_CLASSES_ROOT\ChromeHTML\shell]
[HKEY_CLASSES_ROOT\ChromeHTML\shell\open]
[HKEY_CLASSES_ROOT\ChromeHTML\shell\open\command]
@="\"D:\\Sandbox\\Chrome\\drive\\C\\Program Files (x86)\\Google\\Chrome\\Application\\chrome.exe\" -- \"%1\""
[HKEY_LOCAL_MACHINE\SOFTWARE\Clients\StartMenuInternet\Google Chrome]
@="Google Chrome"
[HKEY_LOCAL_MACHINE\SOFTWARE\Clients\StartMenuInternet\Google Chrome\Capabilities]
"ApplicationDescription"="Google Chrome is a web browser that runs webpages and applications with lightning speed. It's fast, stable, and easy to use. Browse the web more safely with malware and phishing protection built into Google Chrome."
"ApplicationIcon"="D:\\Sandbox\\Chrome\\drive\\C\\Program Files (x86)\\Google\Chrome\\Application\\chrome.exe,0"
"ApplicationName"="Google Chrome"
[HKEY_LOCAL_MACHINE\SOFTWARE\Clients\StartMenuInternet\Google Chrome\Capabilities\FileAssociations]
".htm"="ChromeHTML"
".html"="ChromeHTML"
".shtml"="ChromeHTML"
".xht"="ChromeHTML"
".xhtml"="ChromeHTML"
".webp"="ChromeHTML"
[HKEY_LOCAL_MACHINE\SOFTWARE\Clients\StartMenuInternet\Google Chrome\Capabilities\Startmenu]
"StartMenuInternet"="Google Chrome"
[HKEY_LOCAL_MACHINE\SOFTWARE\Clients\StartMenuInternet\Google Chrome\Capabilities\URLAssociations]
"ftp"="ChromeHTML"
"http"="ChromeHTML"
"https"="ChromeHTML"
"irc"="ChromeHTML"
"mailto"="ChromeHTML"
"mms"="ChromeHTML"
"news"="ChromeHTML"
"nntp"="ChromeHTML"
"sms"="ChromeHTML"
"smsto"="ChromeHTML"
"tel"="ChromeHTML"
"urn"="ChromeHTML"
"webcal"="ChromeHTML"
[HKEY_LOCAL_MACHINE\SOFTWARE\Clients\StartMenuInternet\Google Chrome\DefaultIcon]
@="D:\\Sandbox\\Chrome\\drive\\C\\Program Files (x86)\\Google\Chrome\\Application\\chrome.exe,0"
[HKEY_LOCAL_MACHINE\SOFTWARE\Clients\StartMenuInternet\Google Chrome\InstallInfo]
"ReinstallCommand"="\"D:\\Sandbox\\Chrome\\drive\\C\\Program Files (x86)\\Google\Chrome\\Application\\chrome.exe\" --make-default-browser"
"HideIconsCommand"="\"D:\\Sandbox\\Chrome\\drive\\C\\Program Files (x86)\\Google\Chrome\\Application\\chrome.exe\" --hide-icons"
"ShowIconsCommand"="\"D:\\Sandbox\\Chrome\\drive\\C\\Program Files (x86)\\Google\Chrome\\Application\\chrome.exe\" --show-icons"
"IconsVisible"=dword:00000001
[HKEY_LOCAL_MACHINE\SOFTWARE\Clients\StartMenuInternet\Google Chrome\shell]
[HKEY_LOCAL_MACHINE\SOFTWARE\Clients\StartMenuInternet\Google Chrome\shell\open]
[HKEY_LOCAL_MACHINE\SOFTWARE\Clients\StartMenuInternet\Google Chrome\shell\open\command]
@="\"D:\\Sandbox\\Chrome\\drive\\C\\Program Files (x86)\\Google\Chrome\\Application\\chrome.exe\""
[HKEY_LOCAL_MACHINE\SOFTWARE\RegisteredApplications]
"Google Chrome"="Software\\Clients\\StartMenuInternet\\Google Chrome\\Capabilities"
[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Clients\StartMenuInternet\Google Chrome]
@="Google Chrome"
[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Clients\StartMenuInternet\Google Chrome\Capabilities]
"ApplicationDescription"="Google Chrome is a web browser that runs webpages and applications with lightning speed. It's fast, stable, and easy to use. Browse the web more safely with malware and phishing protection built into Google Chrome."
"ApplicationIcon"="D:\\Sandbox\\Chrome\\drive\\C\\Program Files (x86)\\Google\Chrome\\Application\\chrome.exe,0"
"ApplicationName"="Google Chrome"
[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Clients\StartMenuInternet\Google Chrome\Capabilities\FileAssociations]
".htm"="ChromeHTML"
".html"="ChromeHTML"
".shtml"="ChromeHTML"
".xht"="ChromeHTML"
".xhtml"="ChromeHTML"
".webp"="ChromeHTML"
[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Clients\StartMenuInternet\Google Chrome\Capabilities\Startmenu]
"StartMenuInternet"="Google Chrome"
[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Clients\StartMenuInternet\Google Chrome\Capabilities\URLAssociations]
"ftp"="ChromeHTML"
"http"="ChromeHTML"
"https"="ChromeHTML"
"irc"="ChromeHTML"
"mailto"="ChromeHTML"
"mms"="ChromeHTML"
"news"="ChromeHTML"
"nntp"="ChromeHTML"
"sms"="ChromeHTML"
"smsto"="ChromeHTML"
"tel"="ChromeHTML"
"urn"="ChromeHTML"
"webcal"="ChromeHTML"
[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Clients\StartMenuInternet\Google Chrome\DefaultIcon]
@="D:\\Sandbox\\Chrome\\drive\\C\\Program Files (x86)\\Google\Chrome\\Application\\chrome.exe,0"
[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Clients\StartMenuInternet\Google Chrome\InstallInfo]
"ReinstallCommand"="\"D:\\Sandbox\\Chrome\\drive\\C\\Program Files (x86)\\Google\Chrome\\Application\\chrome.exe\" --make-default-browser"
"HideIconsCommand"="\"D:\\Sandbox\\Chrome\\drive\\C\\Program Files (x86)\\Google\Chrome\\Application\\chrome.exe\" --hide-icons"
"ShowIconsCommand"="\"D:\\Sandbox\\Chrome\\drive\\C\\Program Files (x86)\\Google\Chrome\\Application\\chrome.exe\" --show-icons"
"IconsVisible"=dword:00000001
[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Clients\StartMenuInternet\Google Chrome\shell]
[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Clients\StartMenuInternet\Google Chrome\shell\open]
[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\Clients\StartMenuInternet\Google Chrome\shell\open\command]
@="\"D:\\Sandbox\\Chrome\\drive\\C\\Program Files (x86)\\Google\Chrome\\Application\\chrome.exe\""
[HKEY_LOCAL_MACHINE\SOFTWARE\Wow6432Node\RegisteredApplications]
"Google Chrome"="Software\\Clients\\StartMenuInternet\\Google Chrome\\Capabilities"