How can I replace stock download manager that Google Play uses?
While you intend to replace the stock download manager, but keep access to Google Play Store, since it's your trusted market, before that, you should try to find the reason as to why are you having troubles downloading apps:

Troubleshooting Download Issues


Is the device supported by Google Play Store?

You can confirm this from the Supported Devices List last updated 2012/06/29
Common quick checks:


Restart your phone, then try downloading the app again.
Confirm there is connectivity to either wireless or cellular networks on your device. 

If behind a firewall, ensure that TCP and UDP 5228 ports aren’t blocked (they are required for Google Play).
Clear cache and data of both the Google Play Store app and the Download manager:


Visit Menu > Settings > Applications > Manage applications;
Tab over to "All";
Select the Google Play Store app, and then tap "Clear data" and "Clear cache";
Select Download manager and tap "Clear data" and "Clear cache".



Aditional information for troubleshooting can be found at Google Play: Troubleshoot an issue.



Alternatives



Other App discovery Method

Appbrain, an app discovery service that as per your request still uses Google Play Store as the source for the downloads provide.

It features a great instant download tool (AppBrain App Market), a nice search function, a very clear interface and you can sign-in with your Google account.

One side note: This app still downloads via the stock download manager.



Alternative Markets

Alternative markets have their own download manager. They may be a solution, provided the issue does not lie with your current unofficial ROM:


Amazon Appstore for Android

While an alternative market, I've received a notice from the website stating:


  The Amazon Appstore is only available to customers located in the United States.

GetJar

Launched in 2004, it grows every day, counts 600.000+ apps, and 450.000+ registered developers. As an independent app store, not tied to any particular handset or carrier, it comes as an alternative solution for you.
AndroidPIT

AndroidPIT is a Web 2.0 network and Android blog. Was created to bring together Android experts, enthusiasts and developers. Apps are tested, at least one every day and reviews are published to help users on their decisions.
SlideME

Founded back in 2008, SlideME provides a 'Managed App-Store Service' and builds solutions as per market requirements.