Dictionary translation is "I beg you will favour me with your company―I
request the pleasure of your company."

御来駕被成下度奉懇願候

I can only surmise about ご来駕 noun 懇願候 verb (ask), sourou being a polite form
of suru.

The particles seem to be omitted.

What to make of the 被成下度奉 part?
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grammar classical-japanese
What the grammar/syntax of 御来駕被成下度奉懇願候?
That's a phrase anyone younger than Sherlock Holmes never uses. It's an
example of [候文{そうろうぶん}](https://ja.wikipedia.org/wiki/%E5%80%99%E6%96%87),
which had been official writing style until Edo period but completely died out
around WWII.

Both the wording and orthography are as old as hills (literally). Despite its
look, not much particles are omitted because most of them are embedded in the
form of kanji in this Classical Chinese-inspired notation.

It reads:

> 御{ご}来駕{らいが}成{なし}下{くだ}されたく懇願{こんがん}たてまつり候{そうろう}

where you may notice `被V` and `奉V` stand for common honorific words `V-(さ)れる`
and `V-(し)たてまつる`, and `度` for a conjugated form `たく` of `V-(し)たい`.

Then, if you replace them with present-day vocabulary, it'll be:

> ぜひともお越しくださいますようお願い申し上げます
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The first phrase is nominal. It is composed of 接頭語(prefix)「御」 and
名詞(noun)「来駕」.

[![enter image description
here](https://i.stack.imgur.com/obV1v.png)](https://i.stack.imgur.com/obV1v.png)

The numbers in the right side represent the reading order of 漢字.

The second phrase is composed of a verb(動詞)「[成]{な}す」, a subsidiary
verb(補助動詞)「[下]{くだ}さる」 and an auxiliary verb(助動詞)「たい」.

[![enter image description
here](https://i.stack.imgur.com/O4Jxc.png)](https://i.stack.imgur.com/O4Jxc.png)

The reading order of the second phrase is not simple top-to-down. This kind of
reading order is called 「[返]{かえ}り[読]{よ}み」 which is commonly used in [漢文]{かんぶん}
and [候文]{そうろうぶん}, but almost never used in present-day proses.

「返り読み」 is also used in the last phrase which is composed of 名詞「懇願」,
動詞「[奉]{たてまつ}る」 and 補助動詞「[候]{さうら}ふ」. （The written Japanese word さうらふ = 候ふ is
traditionally pronounced そうろう in spoken Japanese.）

[![enter image description
here](https://i.stack.imgur.com/FVu56.png)](https://i.stack.imgur.com/FVu56.png)

[漢文]{かんぶん} means Classical Chinese writings. However, the most common meaning
of 漢文 in Japan is "Classical Chinese-Japanese word-for-word translation."
Japanese people in some era attempted to translate Classical Chinese writings
into Japanese, trying to maintain the original structure of Classical Chinese
language. So, 漢文 is a writing, written by using 漢字 and Classical Chinese
grammar, and read by using Japanese vocabulary and grammar. 返り読み has been
needed in order to overcome the conflicts between Classical Chinese grammar
and Japanese grammar.

In [候文]{そうろうぶん}, many parts are written by using Japanese grammar, however, a
lot of 漢文 style grammar are also used.

御来駕被成下度奉懇願候 is 候文 and the ways using 被 and 奉 in it are of 漢文 style grammar.

If the order of 漢字 is modified into Japanese grammar and [送]{おく}りがな are added,
it is no longer 漢文 or 候文 (actually some 候文 contain 送りがな though) and called
「[書]{か}き[下]{くだ}し[文]{ぶん}」 or 「[読]{よ}み下し文」. For example, the first Japanese
sentence in broccoli forestさん's answer is 書き下し文.

書き下し文 is always helpful to understand the reading orders of 漢字 in 漢文 and 候文,
especially for people who learned 返り読み in school. (Most teenagers living in
Japan learn 漢文 and 返り読み in school. This 漢文 is translation one, so most
Japanese people don't know how to pronounce them in Chinese.) I posted this
answer in case you don't know the existence of 返り読み in Japanese language.
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