WebSphere Application Server V7 does support Java Platform, Standard Edition (Java SE) 6 (see Specifications and API documentation in the Network Deployment (All operating systems), Version 7.0 Information Center) and it's since the release V8.5 when Java 7 has been supported.
I couldn't find the Java 6 SDK documentation, and could only consult IBM JVM Messages in Java 7 Windows documentation. Alas, I couldn't find the error message in the documentation either.
Since java.lang.UnsupportedClassVersionError is "Thrown when the Java Virtual Machine attempts to read a class file and determines that the major and minor version numbers in the file are not supported.", you ran into an issue of building the application with more recent version of Java than the one supported by the runtime environment, i.e. WebSphere Application Server 7.0.
I may be mistaken, but I think that offset=6 in the message is to let you know what position caused the incompatibility issue to occur. It's irrelevant for you, for me, and for many other people, but some might find it useful, esp. when they generate bytecode themselves.
Run the versionInfo command to find out about the Installed Features of WebSphere Application Server V7, e.g.
C:\IBM\WebSphere\AppServer>.\bin\versionInfo.bat
WVER0010I: Copyright (c) IBM Corporation 2002, 2005, 2008; All rights reserved.
WVER0012I: VersionInfo reporter version 1.15.1.47, dated 10/18/11
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IBM WebSphere Product Installation Status Report
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Report at date and time February 19, 2013 8:07:20 AM EST
Installation
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Product Directory C:\IBM\WebSphere\AppServer
Version Directory C:\IBM\WebSphere\AppServer\properties\version
DTD Directory C:\IBM\WebSphere\AppServer\properties\version\dtd
Log Directory C:\ProgramData\IBM\Installation Manager\logs
Product List
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BPMPC installed
ND installed
WBM installed
Installed Product
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Name IBM Business Process Manager Advanced V8.0
Version 8.0.1.0
ID BPMPC
Build Level 20121102-1733
Build Date 11/2/12
Package com.ibm.bpm.ADV.V80_8.0.1000.20121102_2136
Architecture x86-64 (64 bit)
Installed Features Non-production
Business Process Manager Advanced - Client (always installed)
Optional Languages German
Russian
Korean
Brazilian Portuguese
Italian
French
Hungarian
Simplified Chinese
Spanish
Czech
Traditional Chinese
Japanese
Polish
Romanian
Installed Product
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Name IBM WebSphere Application Server Network Deployment
Version 8.0.0.5
ID ND
Build Level cf051243.01
Build Date 10/22/12
Package com.ibm.websphere.ND.v80_8.0.5.20121022_1902
Architecture x86-64 (64 bit)
Installed Features IBM 64-bit SDK for Java, Version 6
EJBDeploy tool for pre-EJB 3.0 modules
Embeddable EJB container
Sample applications
Stand-alone thin clients and resource adapters
Optional Languages German
Russian
Korean
Brazilian Portuguese
Italian
French
Hungarian
Simplified Chinese
Spanish
Czech
Traditional Chinese
Japanese
Polish
Romanian
Installed Product
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Name IBM Business Monitor
Version 8.0.1.0
ID WBM
Build Level 20121102-1733
Build Date 11/2/12
Package com.ibm.websphere.MON.V80_8.0.1000.20121102_2222
Architecture x86-64 (64 bit)
Optional Languages German
Russian
Korean
Brazilian Portuguese
Italian
French
Hungarian
Simplified Chinese
Spanish
Czech
Traditional Chinese
Japanese
Polish
Romanian
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End Installation Status Report
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