It's possible to install the Bengali Unicode font yourself on an unrooted phone. However, others have found the results aren't that brilliant.
See here for instructions.
The
Opera Mini browser can render all Unicode fonts. To enable it to do this, type
config: in its URL bar, press enter to get to 'Power-User settings', then change 'Use bitmap fonts for complex scripts' to
Yes and Save.
You may already be aware of the
SETT Bengali web browser.
For custom ROMs that will support internet Bengali script, you can investigate those with versions of Android 2.3.6 and above, such as the Cyanogen(-based) ROMs with 2.3.7 and Ice Cream Sandwich (Android 4.0), MIUI ROMs with ICS, and Android Only ROMs with ICS.
See this page for details.