You can use LineJoiner transformer with parameter List Name. But you need to do some additional data processing:
- Use
LengthCalculator to calculate lengths.
- Use
Sorter to sort your lines by length (numeric sort, descending).
- Use
LineJoiner to join lines. The biggest ones first (thanks to step 2). Specify parameter List Name of the LineJoiner - some name for new list attribute.
- Use
ListIndexer with list index 0 - to get the first joined feature (the biggest one) attributes' from the list.
Here is how it will look:

EDIT1:
In order to be able to join only <1m features to bigger ones we need additionally to use:
Tester transformer after LengthCalculator with test: _length < 1
- Use
SpatialFilter (tests to perform parameter: TOUCHES) after Tester to filter only that features which touch features <1m. Then perform line joining on them together with <1m features. Features that don't touch <1m, direct to output (or further processing).
Model:

EDIT2:
If you have <1m line that touches two big ones then they all will be joined together. If such case (<1m feature touches two or more big ones) is possible in your data then you should add some processing of such cases. Take a look at parameters Merge Attributes and Attribute Prefix of transformer SpatialFilter. Using these parameters you should be able to detect such cases and then do some additional filtering.
EDIT3:
Let's solve the case when you have two big lines touching one small (<1m). We need to take only one of the big lines to participate in joining with small one.
In order to be able to perform such filtering we need to mark these two big features with some id from small one they are touching. As stated in EDIT2, it can be done in SpatialFilter transformer using parameters Merge Attributes and Attribute Prefix.
Then use transformer DuplicateRemover to filter such duplicate features (two big) and take only one of them.
Hint: try to use Inspector transformer during creating your model and view intermediate results in any case that is not clear to you. In such way you will better understand how each of the transformers are working. Experiment with your data! :)