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For example: "If you want to practice English, feel free/you are welcome to
message me". How do you carry the nuance of "you are welcome to but don't feel
obliged?" If I said 英語を練習したいならメッセージしてください that would, to my mind, sound too
direct and commanding. But I'm not sure how to make it sound softer.

Edit: As Felipe correctly guessed, the context of use is for HelloTalk, for
people I don't know well or strangers.
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politeness phrase-requests
What's the polite way to invite someone to do something?
I would use メッセージしていいよ。 Literally, "it's alright/okay to message me."

There are a lot of ways to invite, using different levels of formality, but I
think していい(よ) is the most simple, and common.

you could also insert phrases like "any time you like", two common variants
being いつでもいい or 自由{じゆう}に
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I've been using [Hello Talk](https://www.hellotalk.com/) for over two years,
most people that started a conversation with me, said something along the
lines of:

> こんにちは、XXといいます。よかったら、英語を教えてくれませんか？  
>  Hello, my name is XX. If it's ok for you, can you teach me english? (ofc
> you'd switch for japanese here)

But that's more if you are asking for help, in your case sounds like you are
rather offering help or wanting to exchange both languages "equally" right?

In that case I would say:

> こんにちは、XXといいます。一緒に英語と日本語勉強しませんか？よろしくお願いします！  
>  Hello, my name is XX. Let's study english and japanese together? Please
> accept me as your language partner! (It maybe sounds a bit blunt when you
> translate to english, but I think it's totally ok in japanese!)

**Edit:** As pointed by Eric, in case you already know the person and just
want them to feel free to contact/message you:

> 英語を練習したいときに、連絡/メッセージしてくださいね。  
>  Please feel free to contact/message me, **when** you feel like practicing
> english.
>
> 英語を練習したかったら、いつでも連絡してくださいね。  
>  Please fell free to contact me **whenever** you feel like practicing
> english.

Here, I believe the **ね** softens the "request" while still being polite.
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If you want to invite someone to do something politely, you can say ～しませんか?
For example, 一緒に英語を勉強しませんか?、一緒にご飯を食べませんか?

If I were to change your example to a very polite form, I would say
もし英語を練習なさりたいなら、メッセージしていただけませんか?

I found an interesting page about expressions of request in Japanese. I hope
this helps you. <http://web.ydu.edu.tw/~uchiyama/conv/kaiwa_n8.htm>
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[**(お)気軽に**](https://jisho.org/word/%E6%B0%97%E8%BB%BD%E3%81%AB) is a very
handy word that conveys the nuance of "feel free to ～" or "take it easy and
～". Adding お make it sound politer. As an aside, メッセージする as a suru-verb sounds
a little unnatural (or at least a little too colloquial) to me.

Example:

> 英語を練習したいときは **気軽に** メッセージをください。
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